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Here's one way you can think about these problems.
.
Let's begin by talking our way through:
.

Let's start by presuming that the > sign is an equal sign and the equation actually is
.

.
What does the graph of this equation look like? It is a horizontal line through the point
-2 on the y-axis. Does this make sense? What it says is that no matter what value you
select for x, the value of y will be -2.
.
The reality is that the equal sign was only put in to help us picture what is going on
with the graph. Now we can put the > back into the equation. Now we can tell that the
values of y must be greater than -2. This means that y is allowed to be any value
above the line that is the graph. You can shade that entire region, but only the region
that is ABOVE the line. y can be any value in the shaded region. However, y cannot have
the value -2 because y is only allowed to be GREATER than -2. Therefore, y can NOT be
on the line.
.
The next problem says that:
.

.
Like we did before, let's temporarily replace the > sign with an equal sign. This changes
the equation to:
.

.
This is in the slope-intercept form. Maybe you can picture the graph. It crosses the
y-axis at -2 and it slopes up and to the right at a rate of +2. That means for every 1 unit
you move horizontally to the right you go vertically up 2 units. You know that (0,-2) which
is the y-axis intercept is on the graph. You can easily find another point on the graph
by setting y = 0 in the line equation and then solving the equation for the corresponding
value of x:
.

.
When you solve this you find that x = 2 is the answer. Therefore, you know that (2,0) is a
second point on the graph. With the two points (0,-2) and (2,0) plotted you can draw a line
through them and you will have the graph of
.
At this point you should replace the = sign with the > sign to get back to the original
problem. This form tells you that y can only have values ABOVE the graph because y
must be greater than the values in the line. Shade the entire region above the line.
The shaded region is where values of y can be.
Finally, a little more complex (the last problem):
.

.
We can solve this for to make it easier for us to find the region where y is allowed to
exist just as we did before. We want to solve for +y. So let's multiply both sides of
the equation by -1. However, here's an important rule: whenever you multiply or divide both
sides of an inequality by a negative number, you must afterward reverse the direction of
the inequality.
.

.
Do the multiplication by -1 to get:
.

.
But don't forget that you have to reverse the direction of the inequality sign too. When
you do the inequality is now:
.

.
Now you can replace the inequality sign with an equal sign and solve the equation for y
just as you have always done. Begin by adding 4x to both sides to get:
.

.
Divide both sides by 6 to solve for y and get:
.

.
which becomes:
.

.
Graph this equation as you did previously. The slope is (2/3) and the y-axis intercept
is -2.
.
Now replace the = sign with the inequality sign pointing to the right so that the inequality
is now:
.

.
Again, shade the entire region ABOVE the graph of the right side of the equation.
That represents the place where y is allowed. y can NOT be on or below the graphed line.
.
After a little more practice you'll get familiar with this method and you can do things
faster and without thinking about it.
.
Hope this helps you with understanding the basic principles of doing problems such as
these inequalities.

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If two triangles are congruent they will both contain the same angles, and the corresponding
sides will be equal in length.
.
All you need to do to answer this question is to use the same angles in both triangles
but make one of the triangles have longer sides. The triangles will be the same shape but
one will be an enlargement of the other.
.
The way to do this is to draw two horizontal lines on a piece of paper, but make one of the
lines obviously longer than the other. Put angles of 50 degrees at the left end of each of
the two lines you drew. Make the line forming the angle slant up and to the right.
.
Next go to the other ends of the two lines and make 70 degree angles of so the lines
forming the angle slope up and to the left. You should now have triangles that both contain
the same angles, but the corresponding sides of the two triangles are different in length.
.
Hope this helps you understand the problem.

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Two equations are required.
You know that the perimeter of a rectangle is given by adding a length, a width, another length,
and another width to get the perimeter.
Let L represent the Length, W represent the Width, and P represent the Perimeter of the
rectangle. With this notation, the equation for the perimeter of the rectangle is:
.

.
This simplifies to:
.

.
The problem gives the perimeter as 25 meters. Substitute this:
.

.
The problem also says that the length is 1.5 times the width. In equation form, this is:
.

.
Substitute this in place of L in the perimeter equation:
.

.
Multiply out the first term on the right side:
.

.
and combine the two terms on the right side to get:
.

.
Divide both sides by 5 and you find that
.

.
And since the length is 1.5 times as long, the length is:
.

.
Check it.

.
The answer checks.
.
Hope this helps you to understand the problem.

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The exponent of means take the square root of this quantity. In another form your equation is:
.

.
Are you sure this equation is correct? usually represents magnetic flux or possibly
the phase shift in an a-c voltage. Did you mean ... the radian measure of a
circle equivalent to 180 degrees?
.
Also, this is not the equation for the capacitive resistance in an a-c circuit. That equation
is:
.

.
It is also not the equation for the resonant frequency of a series R-L-C circuit.
That equation is:
.

.
I'm a little rusty as you can tell, but I don't recall the equation you provided.

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.
Cube both sides. In the case of the left side that means multiplying it by itself so that
the multiplications are:
.

.
But when you multiply a base term [in this case the base is (x-4)] that has an exponent by
the same base term with other exponents, the product is the same base term, but the new
exponent is the sum of the exponents. For example
.
In the case of taking the cube of

the product will be
.

.
That is it cube of the left side of the original problem. And since we cubed the left
side, we must also cube the right side. 16 cubed is .
.
We have now converted the original problem to:
.

.
Now take the square root of both sides. The left side becomes just . The first
two terms on the right side are and this can be written as .
This gets multiplied by the remaining 16 to become:
.

.
If you now take the square root of this product you get:
.

.
Substituting this in to expression results in:
.

.
Finally, add 4 to both sides to get:
.

.
I hope you are able to follow all the detail. You can check the answer by substituting
this value for x into the original problem so that becomes 64. Then find the
cube root of 64 and square it. Or you may opt to square 64 and take the cube root of
that number. In both cases you should get 16, the same number as on the right side of
the original problem statement.

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Find the equation of the line passing through the points {1/5 , 7} , {-2 , 4}.
.
You don't need to convert the into a "whole number", but maybe it would help you
to change it to its equivalent decimal form which is 0.2 [you can do that conversion
by dividing 5 into 1.0]
If you make that conversion your points are (0.2, 7}, {-2, 4}. Call the first point
(x1, y1) and the second point (x2, y2). By comparing these with the given information
you can see that
x1 = 0.2
x2 = -2
y1 = 7
y2 = 4
The slope of the line joining the two points can be found from the formula:
.

.
Substitute the values for x1, x2, y1, and y2 in the appropriate places of the slope equation
to get

This simplifies to:

and we normally use m to represent the slope. Therefore, you can say m = 1.363636
.
You can now write the equation using:

and substitute for m, y1, and x1 to get:

as the equation.
But if you wanted to stay with the fraction 1/5 as x1, you could write that the
slope m is:

.
You can now work on the value of . Recognize that 2 is the same as
so you can make that substitution to get

Substitute that into the slope equation to get:

Do the division by inverting the denominator and multiplying this inversion times the
numerator:

.
Use the value

for m and substitute it into the equation along with the substitutions
for x1 and y1 to get:
.

.

.
This is an equivalent form of the answer above that uses decimals in the equation.
Study this carefully. Hopefully it will give you some insight into working with fractions
in writing equations.

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I interpret your problem as being asked to simplify:

Let's work on the denominator first. You may recognize is as "a" take away one-half "a".
The result is
If you didn't recognize that, a more general approach would be to say that to combine
fractions you need to have them over a common denominator. How would you combine with
? One has a denominator and the other doesn't. What if you multiplied the
by . [Since is the same as , you are effectively multiplying
by so you are not really changing the . You are just converting
the to .] In this form you now have a common form for the two terms in
the denominator of your problem. That denominator is:

Notice that these two terms have a common denominator of 2 so they can be combined.

Substituting this result into your problem results in your problem now being:

So you are dividing by the fraction }.
One way to look at this is to use the old arithmetic rule ... when you divide by a fraction,
you invert it and then multiply this inverted fraction times the number that you are dividing.
So, using that rule, we invert to get . Then we multiply that by
the that is the original numerator of your problem.

The in the denominator of this product cancels with the in the numerator
to leave just as the simplification of your original problem.
A way that you could do a reality check of your answer is to return to the original
problem and assign a value to . For example, you might say that is 1.
This would make your original problem:

Using "real" numbers may make it a little easier for you to see that the denominator
simplifies to . Then ask yourself, "How many times does
go into ?" A little thought will tell you that there are 2 halves in
. So by using real numbers you have found that the problem reduces to 2. This helps
to confirm that our previous work was correct and that the original problem simplifies
to
Hope this helps.

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.
Do the distributive multiplication on the right side of this equation by multiplying
3 times m and then 3 times -5 to make the equation become:
.

.
On the right side add the two constants -15 and + 7 to get -8. When you do that, the equation
becomes:
.

.
Next you can move the 3m to the left side of the equation by subtracting 3m from both sides.
This subtraction eliminates the 3m on the right side. On the left side you directly
subtract the 3m from the 5m to get 2m. The result is that the equation becomes:
.

.
Then move the + 9 to the right side of the equation by subtracting 9 from both sides.
This eliminates the 9 on the left side. On the right side you combine the -9 and the -8 to get
-17. This makes the equation:
.

.
Finally divide both sides of the equation by 2 to get:
.

.
or doing the division on the right side and expressing the answer in decimal form
.

.
You can check this answer by returning to the original equation and substituting -8.5 for
m.
When you do all the multiplication, addition, and subtraction, you should find that the
left side of the equation equals the right side. If it does, the answer is correct.
Hope this helps you to understand equations and algebraic manipulations.

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.
Notice that 7 can be factored from each of the terms in the numerator so that the problem
becomes:
.

.
In this form it is easier to recognize that the multiplier 7 in the numerator cancels
with the multiplier 7 in the denominator to reduce the problem to:
.

.
Similarly, you can next factor an from every term in the numerator to get:
.

.
Then notice that this multiplier in the numerator cancels with the
multiplier in the denominator. This reduces the problem to:
.

.
Finally, you can factor a from every term in the numerator to get:
.

.
And recognize that the multiplier of the numerator cancels with the
in the denominator to leave you with just:
.

.
This might be the answer you were looking for, but notice also that this can be factored
into:
.

.
and maybe this is the answer form you were looking for.
Hope this helps you to see a form for dividing polynomials that might be useful.

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Bob invested $20,000, part at 14% and part at 13%. If the total interest at the end of
the year is $2,720, how much did he invest at 14%?
It takes two equations to solve this equation. But before beginning, let's define F as the
amount of money invested at 14% and T as the amount invested at 13%.
.
Since the total amount of money invested is $20,000 we can add F and T and set that sum
equal to the $20,000. In equation form this first one of two our equations becomes:
.

.
The problem tells you the interest Bob makes on the amount F is 14% of F. You can write
this as:
.
Interest on F = 0.14*F
.
Similarly the interest Bob makes on the amount T is 13% of T. You can write this as:
.
Interest on T = 0.13*T.
.
These two amounts of interest have to add up to be the total annual interest of $2,720
In equation form this becomes:
.

.
This is the second of the equations that you can use to solve this problem.
There are several ways that you could solve this system of two equations. Let's use the
substitution method. Return to the first equation and see that it can be solved for F
by subtracting T from both sides. If you do that subtraction you get:
.

.
Now substitute that value for F into your second equation to get:
.

.
Multiplying out the first term on the left side results in:
.

.
Add the two terms containing T. That sum is -0.01T and it replaces the two terms,
resulting in the equation:
.

.
Now subtract 2800 from both sides to eliminate the 2800 on the left side. When you subtract
it from the right side (2720 - 2800) the result is -80. Therefore, the equation is reduced to:
.

.
Multiply both sides by -1 to eliminate the negative signs and get:
.

.
Finally divide both sides by 0.01 (or you can multiply both sides by 100 also) and the
equation becomes:
.

.
Since you now know that T = $8,000 and the total amount invested is $20,000 you also know
that the amount invested at 14% (F) has to be $12,000. So the answer you were looking for
is $12,000.
Let's check it:
Do F and T add up to be $20,000?
.

.
That works. Then does 14% of F and 13% of T add up to be $2720?
.
0.14*(12000) + 0.13(8000) = 2720}}}
.
Do the multiplications on the left side:
.

.
The left side does add up to equal the right side, so this works also. The problem is
correct and you know that the answer of $12,000 is correct for the amount invested
at 14%.
.
Hope that this helps you understand the problem a little better, and aids you in figuring
out when you are given two different limits on the problem (in this case the limits of
the total amount of interest and the total amount invested) that you are likely going to
need to solve two equations to get the answer.

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–8x – 5y = –45
To re-write this as a function the basic process is to solve for y and then replace
y with f(x)
.
So let's begin by solving it for y. First you can start by adding 8x to both sides.
On the left side this has the effect of canceling out the -8x. On the right side it makes
a +8x appear. So by adding 8x to both sides the equation becomes:
.

.
Now, to get just y on the left side let's divide all the terms on both sides by the -5
that is the multiplier of y. When you do that you get:
.

.
Performing the division on the right side results in:

.
and this further simplifies to:
.

.
Finish the problem by replacing y with f(x) to make the equation:
.

.
This is the answer you are looking for.
.
Please check the above work. The basic process is OK, but make sure I didn't accidentally
make any math errors.
.
Hope this helps you to understand how to change the standard form of an equation into the
f(x) form.

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y = 5x + 4
Let @ represent the angle theta (theta is a letter in the Greek alphabet. Letters in
the Greek alphabet are used in many math books to represent angles just as letters
in our alphabet such as x and y are often used to represent distances. Don't let this
confuse you and don't worry about it. If you don't like the Greek letter theta, just replace
it with a capital letter from our alphabet ... but to avoid confusion, don't use R, X, or Y.
Use something such as A, B, C ...)
Perform the following steps:
(1) for y in the equation substitute r*sin@
.
r represents the magnitude or length in polar form. It originates from the origin and extends to a point on the coordinate system.
.
(2) for x in the equation substitute r*cos@
.
(3) solve for r
Let's do it.
The given equation is y = 5x + 4
Do the substitution of step 1 to get:
.
r*sin@ = 5x + 4
.
Next do the substitution of step 2 to get:
.
r*sin@ = 5r*cos@ + 4
.
Now solve for r. First subtract 5r*cos@ from both sides. This results in:
.
r*sin@ - 5r*cos@ = 4
.
Factor out the r that is common to both terms:
.
r(sin@ - 5cos@) = 4
.
That's a suitable form of the polar equation that is equivalent to y = 5x + 4.
.
Just for grins, let's try a few points to see if they are equal in both the rectangular
and polar forms. Let's try letting the angle @ be 90 degrees -- or if you prefer pi/2
radians. Plug this value of 2 into the polar form and you get:
r[sin(90) - 5cos(90)] = 4
Now recognize that sin(90) = 1 and cos(90) = 0. Substitute these values to get:
r(1 - 5*0) = 4
which simplifies to:
r(1 - 0) = 4 or simply r = 4 with r starting at the origin (0,0).
Therefore, our polar answer is 4/90 which is magnitude 4 at an angle of 90 degrees.
But think about this. Since we measure angles counter-clockwise relative to the x-axis in
quadrant I, the angle of 90 degrees puts us on the positive y-axis. And since we found
that the magnitude of r was 4, the point determined by the polar form 4/90 is 4 units up
the y-axis from the origin. In rectangular form that point would be (0,4). But does
(0,+4) satisfy the equation we were given? Let's see ... Start with
y = 5x + 4
and substitute 0 for x and 4 for y. When you do you get:
4 = 5*0 + 4
which reduces to 4 = 4. It works! Therefore, we have shown that at least for this
example, the polar form r(sin@ - 5cos@) = 4 and the rectangular form y = 5x + 4 produce
the equal answers of 4/90 and (0,4). Let's try letting @ = 0 degrees in the polar form,
solving for r, then translating to an (x,y) answer and see if it solves our rectangular
form. Something interesting and educational happens here. Begin by substituting
0 degrees into the polar form:
r(sin(0) - 5cos(0)) = 4
Recognize that sin(0) = 0 and cos(0) = 1. Substitute to get:
r(0 - 5*1) = 4
Multiply out and you get -5r = 4 and when you divide both sides by -5 you get
r = -5/4
What does it mean when you get a negative value for r. It means that it actually
runs in the opposite direction (180 degrees away) from what you thought it was. In this
case we thought that r would be at 0 degrees, and the negative sign on the magnitude
tells us that the magnitude is positive, but the angle is actually 180 degrees.
.
Using the convention of angle measurement we now know that r = +4/5 but the 180 degree angle
puts it on the negative x-axis. But in rectangular form, a point on the negative
x-axis has a y value of of zero. Therefore in rectangular form we are at the point
(-4/5, 0). Use these values (x = -4/5 and y = 0) in the equation y = 5x + 4 and see if
they don't satisfy the equation. [Hint: they do!]
By now you should feel pretty comfortable in saying that since the polar and rectangular
equations have produced the same points for some easy comparisons, they are probably
equivalent equations and our polar form is likely to be correct.
Hope this hasn't confused you too much, but if you think about it, maybe you'll start getting
the conversion from rectangular to polar form.

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To determine if the ordered pair is a solution to an equation, just plug the x and y values
of the ordered pair into the equation and see if the left side of the equation still
equals the right side.
So, is the ordered pair (1,-5) a solution to the equation 4x-4 = 9? Note that there is
no "y" term in the equation, so we do not need to be concerned with the value -5 in the
ordered pair. We only need to substitute the x value of the ordered pair into the equation.
So in place of the x in the equation we let x = 1 from the ordered pair. This makes the
equation:
4*(1) - 4 = 9
Simplifying this by doing the multiplication we get:
4 - 4 = 9
which further simplifies to 0 = 9. This obviously is not true. Therefore, the ordered
pair (1, -5) is not a solution to the equation 4x - 4 = 9
In the off chance that you meant the equation to be 4x - 4y = 9, if you substitute
1 for x and -5 for y the equation becomes:
4*(1) - 4*(-5) = 9
After the multiplications are done the equation is:
4 + 20 = 9
This is not true so the ordered pair (1, -5) is not a solution to the equation 4x - 4y = 9 either.
Next, is the ordered pair (1, -5) a solution to the equation 2x + 3y= -13. The same
process is used. From the ordered pair we get the values x = 1 and y = -5. Substitute
those values for x and y in the equation. When they are substituted the equation
becomes:
2*(1) + 3*(-5) = -13
By doing the multiplications this equation becomes:
2 - 15 = -13
which further simplifies to:
-13 = -13
This is true so the ordered point (1, -5) is a solution to the equation 2x + 3y = -13.
Hope these two examples are enough for you to see how to determine if an ordered pair is
a solution to an equation.

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The most important thing to note is that if you are given a line with a slope m, the slope
of a line that is perpendicular to it is
.
You are given the line y = -3x + 1. This is in the form of the slope-intercept equation which
is the form y = mx + b. In this slope-intercept equation m is the slope. By comparing the given equation to
the slope-intercept form you can see that the slope m in the given equation is -3. Therefore,
a line perpendicular to the graph of the given equation is the negative inverse of -3. The
slope of the perpendicular line, therefore, is:
.

.
.
Therefore, the line perpendicular to the graph of y = -3x + 1 has the slope
.
In slope intercept form you can begin to write the equation of this perpendicular:
.

.

The one thing we know about this line is that it goes through the point (0,5). So if you
go to the above,
beginning of our perpendicular equation and substitute 0 for x and 5 for y the equation becomes:
.

.
This reduces to
Substitute 5 for b in the equation for the perpendicular line. With this substitution
the equation becomes:

Trust me, this is correct. However, it does not match any of the answers listed. Somebody
made a mistake. If you have doubts, graph the given equation and the above equation
for the perpendicular line and you will see they are perpendicular, and the perpendicular
line goes through (0,5).

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I think what you are asking is, "If 4 times a number is added to 13, the result is 29. What
is the number?"
You can use x to represent the unknown number and then write this problem in equation form as:

Subtract 13 from both sides of the equation to eliminate it from the left side of the equation.
The result of this subtraction is:

or just:

Now divide both sides by 4 so that you just have x on the left side:

After this division the left side is just x and the right side is 4. The answer is that the
missing number is 4.
Check: Does 29 = 13+(4*4). It does because 13 + 4*4 = 13 + 16 = 29.
Hope this helps you see your way through this problem.

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.............
Inside the parentheses, divide 0.052 by 4. The answer to that is 0.013. So inside the
parentheses you can replace by 0.013. This changes the problem to:

Now add the two terms in the parentheses to get a sum of 1.013. This simplifies the problem
to:

Raising a number to the 4th power is defined as multiplying it by itself a total of 4 times.
So is the same as . Calculator time. The product
is 1.053022817.
Note: some calculators have a function key that looks like or something
similar. You could use this key as follows: enter 1.013, press the key, and
then press 4, and finally press the = key. You should get 1.053022817 this way also.
Now all you have to do to get the answer is to multiply 3700 by 1.053022817. If you do you
should have the final answer as: 3896.184421
Hope this helps.

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Presume that the number of shots that Phill took is represented by S
If he made 0.75 of them (75% or 3 out of every 4) then 0.75 times S equals the 12 shots
that he made.
So you can write the equation:
0.75*S = 12
Solve this equation by dividing both sides by 0.75

On the left side of this equation the 0.75 in the denominator cancels with the 0.75 in
the numerator. So all that remains on the left side is S. On the right side if you divide
12 by 0.75 the answer is 16. (You should be able to do this with paper and pencil, but
you can use a calculator if necessary.) So Phill took a total of 16 shots.
Check by saying that Phill was making 3 of every 4 shots he took. In the first 4 shots he
probably made 3. In his second 4 shots he made another 3. In the third 4 shots he made 3
more. And in his next 4 shots he made another 3. If you add all these up you find that
he made a total of 12 free throws in the 16 shots that he took. That seems to work.

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Write the equation for a graph that has a slope = 2 and a y-intercept of 0
.....
The definition of the slope-intercept form of an equation is y = mx + b
in which
m is the slope and b is the y-intercept.
All you have to do now is substitute the given values of +2 for m and 0 for +b. This makes
the equation that you are looking for: y = +2x + 0 which reduces to just y = +2x
........
Write the equation for a graph that has a slope of 6 and passes through (-4,7).
You can again use the slope-intercept form:
y = mx + b
Substitute the given slope for m to get:
y = 6x + b
You also given that when x = -4 and y = +7 it satisfies the equation. Substituting
these two values into the equation result in:
7 = (6)*(-4) + b
After the multiplication you have:
7 = -24 + b
Solve this equation for b by adding 24 to both sides to get:
31 = 0 + b
which becomes just b = +31
Return to the equation y = 6x + b and substitute +31 for b. When you do the equation
that has a slope of 6 and goes through the point (-4,+7) becomes:
y = 6x + 31
Hope this gives you a little insight into graphing equations of the slope-intercept
form.

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Set y equal to the function you were given. This results in the equation:

Now think about the general structure of a graph. If you have a point on the Y-axis,
what is the value of X for that point. Every point on the Y-axis has a corresponding
X value of zero!!! Think of +3 on the Y-axis. This would be the point (0,+3). Similarly
the point -5 on the Y-axis would be (0, -5) and so on.
So for the above equation we can simply set x equal to zero to find the corresponding
value of y that satisfies the equation. Start with:

Plug in zero for x. This causes the terms that contain an x to be zero. What you are
left with is:
which simplifies to just
The Y-intercept occurs at y = -8 or at the point (0,-8). It's harder to explain than it
is to just set x equal to zero and determine what is left on the right side of the equation.
Hope this helps you understand how to find Y intercepts.

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One method commonly taught is called FOIL. It means that for problems involving 4 terms that
are in two sets of parentheses such as these, multiply the FIRST terms in each set of
parentheses.
Next multiply the OUTSIDE terms of the four terms and add the result to the product of
the first terms.
Then multiply the INSIDE terms of the four terms and add the result to the previous answers.
Finally, multiply the LAST terms in each set of parentheses and add that result to the previous
answers.
Let's work the problem:
Multiply the FIRST terms (the F in FOIL). In each set of parentheses for this problem
the first term is x. So multiply x times x to get .
Next multiply the OUTSIDE terms (the O in FOIL). The outside terms of the four terms
is x from the first set of parentheses and +9 from the second set of parentheses.
The product of these two terms is +9x. Add this to the previous answer and you
now have

Then multiply the two INSIDE terms (the I in FOIL) of the four terms. The inside terms
are the two middle terms of the four terms. These are: -10 from the first set of parentheses
and +x from the second set of parentheses. The product of these two terms is -10x.
Add this to the previous answer and you have .
Finally, multiply the LAST terms in each set of parentheses (the L in FOIL). The last term
in the first set of parentheses is -10 and the last term in the second set of parentheses
is +9. Multiplying these two results in -90. Add this to the previous result to get
.
Notice in the answer that the terms + 9x - 10x can be added to get -x. When you do that,
the answer simplifies to the final answer of:

Hope this helps you to understand problems of this form. Just FOIL them!

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the rectangular coordinates of a particular point are x=7, y=-24, find the polar coordinates...
Note that since the value of y is negative and the value of x is positive. This puts
the point in quadrant IV.
The magnitude of the radius of is the hypotenuse of a right triangle with legs of length
7 and 24. Use the Pythagorean theorem to find this hypotenuse (H).


That gives us Magnitude. All we need now is the angle. The ratio of the angle is legs
of the triangle are:

This meets the definition of the tangent of the angle whose opposite side is -24 and whose
adjacent side is +7. Notice how this falls into quadrant IV = Since this ratio is the
tangent, by applying arctangent on a calculator, we find that the angle is -73.73979 degrees.
The polar form can be written in two ways:
25/-73.73979
which means a magnitude of 25 units with an angle found by rotating the magnitude
clockwise from positive portion of the x-axis by the amount 73.73979 degrees.
The angle can also be established by rotating (360-73.73979 or 286.26021 degrees)
counter-clockwise from the positive portion of the x-axis. This would be written as:
25/286.26021
I hope this helps you to understand the relationship between the Cartesian and polar representations
of points and specifically how to change from rectangular to polar form.
You may also find it helpful to make a rough sketch of the rectangular point (7,-24) and
draw the magnitude (radius) from the origin to this point. Measure out 7 on the positive
x-axis and the vertical distance from the point (7,0) to the point (7, -24). This will
help you to see the right triangle (0,0) to (7,0), from (7,0) to (7, -24), and then from
(7, -24) to (0,0). The two points (7, -24) to (0,0) are the ends of the radius and the
Magnitude of this line can also be found using the formula for the distance between two points.
the tangent of the angle is the ratio of the length of the line (7,0) to (7, -24) over
the length of the line from (0,0) to (7,0). Such a rough sketch will help you to visualize
the problem.

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Multiply both the numerator and the denominator by

The denominator multiplies out to become 7 and the numerator becomes
So the answer is
The reason for simplifying the original problem, is that convention says to never leave a
radical in the denominator. Get rid of it by multiplying it by itself.
Hope this helps you.

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Recognize that by definition and
From this we can develop the sequence:









Note that this series is i, -1, -i, 1, i, -1, -i, 1, i, ....
We can use this table along with some rules of exponents to simplify the terms in the problem.
Let's try to simplify . We use one of the laws of exponents to rewrite
the this as . From the above table you can see that
Substitute this to get . We can further use a law of exponents
to factor this term into . But so:

So .
We can use this result to find . Note that . Substitute
-1 for to get .
Then use this result and procedure to find that:

Repeat this process to find that:

Substitute these four results into the original problem:

The substitution results in:

The answer to your problem is 2i.
Hope this helps.

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Addition
(1) When adding two numbers having the same signs, add the numbers and prefix the answer
with the common sign.
Examples:
(+3) + (+6) = +9 and
(-4) + (-2) = -6
(2) When adding two numbers of unlike signs, subtract the smaller number from the larger number,
and give the answer the sign of the larger number. Examples: (+9) + (-6) = +3 and
(-9) + (+6) = -3
Next:
Subtraction: Change the sign of the quantity being subtracted. Then follow the rule for addition.
Examples:
(+5) - (+3) =?? Keep the +5 but change the +3 to -3. Then follow addition rule (2) signs by
subtracting 3 from 5
and prefixing the answer 2 with the same sign as the bigger number, that is the sign of
the 5 which is +. Therefore, the answer is + 2
Another example. (+5) - (-9) =? Change the sign of the number being subtracted and
follow the rules of addition. When you change the sign the -9 becomes + 9 which has the
same sign as the +5. Therefore, follow addition rule (1). Add the two numbers to get 14
and add the common sign which is +. The answer is +14.
Another example. (-9) - (-5). Change the sign of the -5 so it becomes +5 and add it to
-9. Use addition rule 2 because the two numbers (+5 and -9) have different signs.
Subtract the two numbers (9-5) and prefix the sign of the larger number (9 which has a
minus sign). The answer is -4.
Another example: (-8) - (+6) = ? Change the sign of the +6 so that it becomes -6. Then
add the -8 to the -6 by following addition rule 1. Since the numbers have the same sign, add
the numbers (8+6) to get 14 and prefix the common sign of the two which is the negative sign. The answer is -14.
Multiplication.
(1) Multiply the two numbers. If they both have the same sign, the answer will be positive.
Examples: (+4) * (+5) = +20 and (-4)* (5) = +20. [That's correct, both answers are + 20
because in both examples the signs of the numbers being multiplied are the same ... in
the first example they are both + and in the second case they are both -.]
(2) Multiply the two numbers. If they have different signs, the answer will be minus.
Examples: (+4) * (-5) = -20 and (-4) * (+5) = -20
Division.
The same general idea as multiplication, except this time you divide the two numbers and
use the rule that if both the numbers have the same sign the answer is +, but if the two
numbers have opposite signs the answer will be -.
Example: but and but

Hope this helps you to get a grasp of the basic rules. The rest involves lots of practice
until they become automatic to you.

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The quadratic formula says that if you are given an equation in the form:

then the values of x that satisfy this equation are given by the two equations:
and

So the first thing we need to do is to get the problem you are given into the form of the
quadratic formula. Note that the quadratic formula has a zero on the right side. However,
your equation has a 3 on the right. So you need to subtract 3 from the right side of
your equation, but if you do you must also subtract 3 from the left side.
When you do the subtraction of 3 from both sides, the left and right sides become:

That's more like it. You can now compare the left side of this equation with the left
side of the quadratic formula. Note that "a" in the quadratic formula is the multiplier
of the term. The multiplier of in your equation is 5. Therefore,
you can deduce that . Similarly, b in the quadratic formula is the multiplier
of the x term. The multiplier of the x term in your equation is +1. This time you can
deduce that . Finally c in the quadratic formula is the constant term on the left,
side. In your problem the constant term is -3. Therefore, c = -3.
Now you have all the information you need to find the values for x that make the equation
true. All that you have to do is to plug in the appropriate values for a, b, and c.
The first value of x is determined from the equation:

After the substitutions this equation becomes:

This simplifies to:

which further simplifies to:

By comparing this answer for x with the second value of x you can notice that the only
difference is the sign between the first term and the square root term. In the first
answer the sign is + and we calculated that. The calculations for the second answer is:

which has the minus sign between the terms.
With a little thought you will be able to see that the answer is the same as answer D.
[The denominator 10 in our answers is common to both terms so that both numerators
can be combined over this common denominator.]
Hope this helps you understand the quadratic formula a little better.

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A roof rises 9.75 ft over a horizontal distance of 17.24 ft. What is the slope of the roof to the nearest hundredth?
Slope is defined as the amount the roof goes up (vertical) divided by the corresponding
distance that it goes horizontally.
Since the roof goes up 9.75 feet in 17.24 feet of horizontal distance the slope (call it
S) is:

Calculator time. Do the division to find the answer is 0.56554 which rounds to 0.57
to the nearest hundredth.
Architects generally do not identify the Slope of a roof this way. They express it in
terms of the pitch ... how many inches does the roof go up per inches of horizontal
run. For example, a "4 in 12" pitch means the roof goes up 4 inches for each 12 inches it
goes horizontally. To convert this slope to architect's pitch we could write:

where V is the vertical distance and 12 is the number of inches in a foot. By substituting
0.57 for S, the equation becomes:

Solve this by multiplying both sides by 12 to get:

So in this case the Architect would say the pitch of the roof is 6.84 in 12 meaning
that the roof rises 6.84 inches for each 12 inches of horizontal distance.
Recognize that this would drive a carpenter crazy. A carpenter would know how to build
a 6 in 12 pitched roof or a 7 in 12 pitched roof, but the mathematics of being a 6.84
in 12 pitched roof would drive most of them into fits of laughter. The architect
would generally specify a 7 in 12 pitch for this problem. But then mathematicians
never seem to worry about the practicality of their answers ... sigh ...
Sorry for
the "extra" information, but it's a slice of real life.

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Joyce makes $6000 more per year than her husband does. Joyce saves 10% of her income for retirement and her husband saves 6%. If altogether they save $5400 per year, then how much does each of them earn per year?
Suppose we let H represent the amount that her husband earns in a year and J represent the
amount that Joyce earns.
The first sentence tells you that J is $6000 more than H. So if we take away $6000 from
J the amount left should be H. In equation form this is:

For later convenience, let's rearrange that by first adding $6000 to both sides and then
subtracting H from both sides to get:

Now let's look at the second part of the information given in the problem.
Joyce saves 10% of her income. That is she saves 10% of J or in decimal form she saves
. Her husband saves 6% of his income or in decimal form he saves .
Added together, these two amounts of savings totals $5400. In equation form this becomes:

You now have 2 linear equations that can be solved simultaneously:
and

One way to do it is to multiply the entire bottom equation by -10 so that it becomes:

If you add this equation to the first equation, the J and the -J cancel and the resulting
equation becomes:

Calculator time. Divide both sides by -1.6 to get

And since Joyce makes $6000 more than that, you know she makes:

Check by seeing if 10% of Joyce's salary plus 6% of her husband's salary adds up to be $5400.
Hope this helps you to see that problems of this sort require 2 equations to solve, and
that they must be solved as simultaneous linear equations.

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The slope-intercept form is y = mx + b in which m and b are two constants.
All you have to do is to rearrange the given equation so that it is in the form:
y = mx + b.
The first thing to notice is that in the equation you were given, the term containing x
is on the left side, but in the slope-intercept form it is on the right side. So to move
the term containing the x to the right side, you can add x to both sides. Adding x
on the left side has the effect of canceling out the -x. And adding x on the right
side puts an x on the right side. The resulting equation is now:

When you compare this equation with the slope-intercept form, you should see that the
slope-intercept form has only y on the left side. But your equation now has 5y on the left
side. You need to divide the left side of your equation by 5 so that it just becomes
y. But if you divide the left side by 5, you must also divide all the terms on the right
side by 5. The division is:

When you do the division you get:

Now you have it exactly in the slope-intercept form. You have just y on the left
side and on the right side you have and
Hope this helps.

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x + y = 3
x + y = –1
This is a "trick" question. Let's convert the two equations to the slope-intercept form:
y=mx+b
In this form the multiplier of x (which is m) is the slope of the graph and b is the value
of y at which the graph crosses the y-axis.
We can convert the top equation to this form by subtracting x from both sides. The
resulting equation is:
y = -x + 3
Compare this equation with the slope-intercept form. Note that the multiplier of x is -1,
so the slope of the graph is -1. The point at which the graph crosses the y-axis
is b which in this case is plus 3.
Now let's re-arrange the second equation into the same slope-intercept form. We do that
by subtracting x from both sides to get:
y = -x - 1
Note that by comparing this equation with the slope-intercept form we again find that
the slope (the multiplier of x) is -1, but this time the point at which the graph
crosses the y-axis (that is the point b) is -1.
What does that tell us? Because the two graphs of these equations have the same slope
they are parallel!!! The only difference is that one line is higher up (crossing the
y-axis at y=3) than the other line (crossing the y-axis at y=-1).
If the pair of equations has a common solution, the two graphs must intersect each other
at that common point. Since the two graphs are parallel in this case, there never
intersect. Therefore, there is no point common to the two graphs. There are no
values
for x and y that will satisfy both equations. Somebody tried to trick you ... or at least
wanted you to think about the situation.
And if you think about it, by looking at the original equations you might have questioned
how in one case x added to y could give you 3 as an answer and in the very next equation
the same value for x added to the same value for y could give you -1 as an answer.
It would have been a clue that something wasn't right.
Hope this helps your understanding of pairs of linear equations.

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2x – 4y = 7
4x + 2y = 9
Note that you can't solve this pair of equations by just adding them together because
there isn't a term in the first equation that is equal to a corresponding term in the
second equation but that has the opposite sign so that when they are added they cancel.
You must first do something to one of the equations that make a term in each equation
have the same term but also have opposite signs. For example, in this problem, what would
happen if you multiplied all the terms of the bottom equation by 2. The pair of equations
would then be:
2x – 4y = 7
8x + 4y = 18
Now you can see that the term 4y appears in both equations, but it has a minus sign in
the first equation and a plus sign in the second equation. Now if you add the two
equations vertically (this satisfies the need to solve it by addition) the 4y terms
cancel and the equation you end up with is:
10x = 25
Dividing both sides by 10 results in:
x = 2.5
You can now solve for y by substituting 2.5 for x in either of the two original equations.
Let's use the first equation.
2x-4y = 7
Put 2.5 in place of x to get:
2(2.5) - 4y = 7
Multiply out 2(2.5) and the equation becomes:
5 - 4y = 7
Subtract 5 from both sides and the equation reduces to:
-4y = 2
Finally divide both sides by -4 and you are left with:
y = -0.5
That solves the problem ... x = 2.5 and y=-0.5 are the answers.
Is that the only way to solve this problem by addition? Nope. Way back at the beginning
we could have multiplied the top equation by -2. This would have made the pair of
equations:
-4x +8y = -14
4x + 2y = 9
You can now solve this pair by addition because the minus 4x will cancel with the plus 4x.
Try it. You should get the same answers for x and y as we did above.

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Your problem is:

The negative exponent means that the term raised to that negative value can be changed
into a fraction having 1 as the numerator and the term with a positive exponent as the
denominator. Easier to show by an example than to explain it.
Think "I can change into by making the denominator
of a fraction that has 1 as its numerator." When you do that your problem becomes:

Then you can eliminate the denominator if you multiply both sides by
If you do that multiplication on the left side you have a fraction that has
in both the numerator and denominator. This is equivalent to 1. On the right side the multiplication results in and the problem becomes:


Dividing both sides by 9 gives:

Now just take the square root of both sides. Notice that because
. Also notice that both the positive and negative forms
of are needed because when either is squared they produce a positive value
of
The two answers to this problem are:
and
You can check this by putting each of these answers (positive and negative) one at a time
into the equation:

and see that with these values the left side of the equation equals 9.