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(y - 8)/2 = (y + 10)/5
Clear the fractions by multiplying both sides of this equation by 10 (the LCM of 2 and 5).
5*(y - 8) = 2*(y + 10)
Use the distributive property to get rid of the parentheses.
5*y - 40 = 2*y + 20
Combine like terms; subtract 2*y from both sides and add 40 to both sides.
5*y - 2*y = 20 + 40
3*y = 60
Solve for y by dividing both sides by 3.
y = 60/3
y = 20
Check the answer by substituting 20 for y in the original equation, and then simplify to see if you end up with a true statement.
(20 - 8)/2 = (20 + 10)/5
12/2 = 30/5
6 = 6
Our solutions checks out.
~ Mark

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3*a + b = 8 is an equation.
We cannot solve this equation because there are two variables (a and b).
Did you get any instructions for this equation?
~ Mark

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There are many different forms for writing the equation of a line. For this problem, we need to know two of them:
y = m*x + b
This is the "slope-intercept" form where m is the slope and b is the y-intercept. They call it the slope-intercept form because you can see at a glance what the slope and y-intercept are when looking at the equation.
For example, y = -7*x + 13
We can see that this line has a slope of -7 and crosses the y-axis at (0,13).
The other form that we need to know is the "point-slope" form. We use this form when we know the slope of the line and the coordinates of one point on the line.
This is the information that you've been given. The slope is -2, and the known point is (-3, -5).
If the known point is (a, b), and the slope is m, then the point-slope form is:
y - b = m*(x - a)
So, substituting the known coordinates for a and b, and the slope, we get:
y - (-5) = (-2)*(x - [-3])
y + 5 = -2*(x + 3)
y + 5 = -2*x - 6
y = -2*x - 6 - 5
y = -2*x - 11
So, as you see, by solving the point-slope form for y, we get the slope-intercept equation of the line.
This line has a slope of -2, and it crosses the y-axis at (0, -11).
~ Mark

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(^2-3)(^2+3)
This is not an equation. An equation always contains an equals sign.
Additionally, these expressions do not make sense because you cannot begin with the ^symbol.
If you mean:
(x^2 - 3)*(x^2 + 3)
and the instructions are to simplify, then this is the factorization of a "difference of squares" -- which should be discussed in your text.
The pattern is:
(a^2 - b^2) = (a + b)*(a - b)
In your problem, a = x^2 and b = 3.
Squaring both and taking their difference gives us:
x^4 - 9
~ Mark

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Consider these two rational numbers: 0 and 1. (We can think of them as rational numbers because they can be expressed as 0/1 and 1/1.
Between them is the rational number 1/10.
Between them is the rational number 1/100.
And so it goes; all of these numbers are between 0/1 and 1/1:
1/1000
1/10000
1/100000
1/1000000
Since this goes on forever, we can say that there is a rational number between every two rational numbers.
~ Mark

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To solve this problem algebraically, we need to know a property of logarithms.
ln(a*b) = ln(a) + ln(b)
In other words, if we take the natural log of a product, that is the same as adding the natural logs of the factors.
We need to find the value of x such that f(x) = g(x).
17.5*e^(0.0006*x) = 14.1*e^(0.0187*x)
Take the natural log of both sides.
ln[17.5*e^(0.0006*x)] = ln[14.1*e^(0.0187*x)]
Now use the property described above to write each side as a sum of natural logs.
Remember that ln(e^x) = x, so ln[e^(0.0006*x)] = 0.0006*x.
ln(17.5) + 0.0006*x = ln(14.1) + 0.0187*x
Group like terms; subtract 0.0006*x and ln(14.1) from both sides.
0.0181*x = ln(17.5) - ln(14.1)
Use a calculator to evaluate the expression on the right side.
0.0181*x = 0.2160
Solve for x.
x = 11.9
Adding 12 years to 1998 gives the desired result.
~ Mark

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If we add 1/10 to 5, we get 5 + 1/10 which equals 51/10 (a rational number with a decimal representation of 5.1).
If we add 1/100, we get 501/100 which equals 5.01.
If we add 1/1000, we get 5001/1000 which equals 5.001.
We can continue this forever:
50001/10000 = 5.0001
500001/100000 = 5.00001
5000001/1000000 = 5.00001
Since this process can go on forever, and each time we are adding a smaller and smaller amount to 5, there are an infinite number of rational numbers between 5 and 14.
~ Mark

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What blank?
~ Mark

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Hello There:
What are you trying to graph?
~ Mark

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x^2 - 14*x + 49 is factorable if we can find two numbers that multiply to make 49 and add to make -7.
With a little thought, you should realize that these two numbers are -7 and -7.
(-7)*(-7) = 49
-7 - 7 = -14
So, the factorization is:
(x - 7)*(x - 7) = (x - 7)^2
~ Mark

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Off the top of my head, I can think of five different ways to solve this problem. I don't know which method(s) you've studied, so I have to guess.
I'll use the "substitution method".
The strategy in this method is to solve one of the equations for one of the variables. You'll end up with an expression that contains the other variable.
Substitute this expression into the other equation, and that will give you an equation that contains only one variable. Solve it in the normal way.
Once you have the value of that variable, substitute it into the expression you got from solving the first equation.
Here goes:
I choose to solve the first equation for 'y'.
x - 4*y = 15
-4*y = 15 - x
y = (15 - x)/(-4)
Now I substitute this expression for y in the second equation.
2*x + 5*y = 4
2*x + 5 = 4
I'll clear the fraction by multiplying both sides by -4.
-8*x + 5*(15 - x) = -16
-8*x + 75 - 5*x = -16
-13*x = -91
x = -91/(-13)
x = 7
Now that we know x is 7, we substitute it for x in our formula for y above.
y = (15 - x)/(-4)
y = (15 - 7)/(-4)
y = 8/(-4)
y = -2
The solution is x = 7 and y = -2.
You can verify this solution by substituting 7 for x and -2 for y into both of the original equations to see that it yields true statements.
~ Mark

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Hello There:
To make y a function of x, you need to have y all by itself on one side of the equals sign and then an expression containing x on the other side.
So, your example of y = x - 3 is already in the correct form.
~ Mark

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Hello There:
I just responded to this problem. See question 20587.
~ Mark

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Hello There:
I just responded to this question.
See problem 20588.
~ Mark

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To find the cost of one roll of tape, we divide the given price by 2 because the given price is for twice the number of rolls that we want.
$1.28/2 = $0.64
----------------
To find the cost of two note pads, we first divide the given price by 3 because we want to know how much one note pad costs.
$0.87/3 = $0.29
Each note pad is 29 cents. Since we want to note pads, we double this amount.
$0.29*2 = $0.58
----------------
One highlighter costs $1.59. We want two of them, so we double this amount.
$1.59*2 = $3.18
----------------
We want two boxes of ink catridges, and we are told that two boxes cost $1.98.
----------------
Let's total the amounts:
$0.64 + $0.58 + $3.18 + $1.98 = $6.38
We would pay $6.38 for these seven items.
~ Mark

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In one minute, Paul types 60 words and Jennifer types 80 words.
80 - 60 = 20
Paul types 20 words per minute slower than Jennifer.
~ Mark

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This question wants to know what 5.25% of 200 is.
Remember, the percent sign (%) means divide by one hundred. Dividing a decimal number by one hundred is easy; we just move the decimal point two places to the left.
So, 5.25% = 0.0525
To find the percentage, we multiply this 'decimal' form of the percent by 200.
0.0525*200 = 10.5
The 200-pound piece of alloy contains 10-1/2 pounds of copper.
~ Mark

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A square yard is a measure of area. To answer this question, we need to know two things.
1) How to calculate the total area carpeted (this will be the number of square feet).
2) How to convert square feet into square yards.
To find the area, we use the simple formula:
A = Length times Width
A = 12*16
A = 192
The room is carpeted with 192 square feet of carpet.
A square yard is a square that measures 3 feet by 3 feet because there are 3 feet in one yard. Therefore:
1 square yard = 3*3 square feet
1 square yard = 9 square feet
To find the square yards, we divide 192 square feet by 9 to find out how many squares there are that measure 3 feet by 3 feet.
192/3 = 21-1/3
There are 21 and 1/3rd square yards of carpet. Each of these cost $18.
18*21-1/3 = 384
It costs $384 to carpet this room.
~ Mark

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We need to know two things to answer this question.
1) How to calculate the volume of the tank in cubic inches (we use inches because that's the unit of measurement we are given).
2) Finding out how many cubic inches there are in one gallon of water.
Let's do #1 first. The volume of a rectangular container is Length times Width times Height.
V = 18*10*9
V = 1620
The fish tank holds 1,620 cubic inches of water.
Now we do #2. I did not know how to convert gallons into cubic inches, so I went to the following site:
http://www.onlineconversion.com/volume.htm
Here, I found out that 231 cubic inches equals 1 U.S. liquid gallon.
Now it's just a matter of finding out how many 231 cubic inch amounts there are in the fish tank.
1620/231 = 7.01
The fish tank holds 7 gallons of water.
~ Mark

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Let x = the number of candles produced and sold.
The costs are modeled by this equation:
C = 200 + 1.5*x
because each x costs $1.50, and there is the startup costs (which is fixed one time only) of $200.
The revenue is modeled by this equation:
R = 4*x
because they collect $4 for each x.
To break even means that the costs equal the revenue (i.e., there is no profit, but there is no loss either).
We find this number of candles by setting the costs equal to the revenue.
200 + 1.5*x = 4*x
Subtract 1.5*x from both sides so that we can combine the x-terms together on the right side.
200 = 4*x - 1.5*x
200 = 2.5*x
Solve for x by dividing both sides by 2.5:
x = 200/2.5
x = 80
~ Mark

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Hello There:
Here is the information that you need to answer these questions.
An equation of the form x = c, where c is a real number, has a graph that's a vertical line.
Your two equations x = -2 and x = 3 are vertical lines that intersect the x-axis at (-2, 0) and (3, 0).
An equation of the form y = c, where c is a real number, has a graph that's a horizontal line.
Your equation y = -7 is a horizontal line that intersects the y-axis at (0, -7)
When two lines have the same slope, they are parallel.
When two lines have slopes that are negative reciprocals of each other, they aer perpendicular.
Solving for y shows us the slope.
This equation is already solved for y:
y = 2*x - 1
Solving this equation: y/2 = x + 3 gives us:
y = 2*x + 6
Both of these lines have slope 2, so they are parallel.
Solving 2*y = -x leads to: y = -(1/2)*x
This slope is -1/2; so it's the negative reciprocal of 2 (the slope of the parallel lines).
So, 2*y = -x is the line that is perpendicular to y/2 = x + 3 and y = 2*x - 1.
~ Mark

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Solving an equation that contains one radical term requires that you first isolate the radical on one side of the equals sign and then square both sides.
sqrt(d^2 - 19) - 2*d + 11 = 0
Add 2*d and subtract 11 from both sides to isolate the radical.
sqrt(d^2 - 19) = 2*d - 11
Next, square both sides.
d^2 - 19 = 4*d^2 - 44*d + 121
This is a quadratic equation. Let's put it into general form.
3*d^2 - 44*d + 140 = 0
Hopefully, you know how to solve this using the quadratic formula.
Do this gives us two candidates for the solution:
d = 14/3 and d = 10
NOTE: It is VERY IMPORTANT TO CHECK YOUR ANSWERS anytime you square both sides of an equation in the process of solving it. Squaring both sides can introduce solutions to the quadratic equation which do not work in the original equation.
Checking with the original equation, we find that 14/3 is not a solution.
Therefore, the answer is d = 10.
~ Mark

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Hello There:
I interpret your equation as:
(-12*m)/(-3) = -12
This can be written differently.
(-12/-3)*m = -12
Dividing negative 12 by negative 3 gives positive 4.
4*m = -12
Solve for m by multiplying each side by 1/4 (this is the same thing as dividing both sides by 4).
m = -3
~ Mark

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32000/n = 8000
You did not ask any questions, so I'm going to guess that you need help solving for n. In other words, we need to find out which number, when divided into 32000, results in 8000.
If we multiply the left side by n, then we will have n/n which is equal to one. Of course, if we multiply one side of an equation by some amount, we must always multiply the other side as well.
This is a basic rule of algebra; whatever you do to one side of an equation, you must do to the other side.
We multiply both sides by n.
32000*n/n = 8000*n
32000 = 8000*n
The last step is to divide both sides by 8000. This will isolate the n on the right side, which is what we want.
32000/8000 = (8000/8000)*n
4 = n
~ Mark

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This problem asks for two unknown amounts, so we need to come up with expressions to represent them. Since we are told that the relationship between the sales of thermometers and hot-water bottles is that the number of thermometers sold is seven times the number of hot-water bottles, we can assign the following expressions for the number of each.
x = the number of hot-water bottles sold
7*x = the number of thermometers sold
The amount of money that Wal-Mart collects from the sale of each is expressed as PRICE times QUANTITY.
So, the money collected for selling x number of hot-water bottles is 6*x.
The money collected for selling 7*x thermometers is 2*(7*x) which is 14*x
We can now write an equation because we know what the total amount of money collected is:
6*x + 14*x = 1200
Combine like terms on the left side:
20*x = 1200
Solve for x by dividing both sides by 20:
x = 1200/20 = 60
Remember what x stands for. This result tells us that 60 hot-water bottles were sold.
7*x = 420
There were 420 thermometers sold.
Let's check these results.
$2*420 + $6*60 = $840 + $360 = $1,200
It checks out; our answer is correct.
~ Mark

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We have three unknowns (the sides of the triangle), so we need to assign three variable names for these unknowns and then come up with three equations to solve the whole system.
Let x = the first angle
Let y = the second angle
Let z = the third angle
From the problem information, we can write:
y = 6*x
and
z = y + x^2 - 18
Of course, we can also write:
x + y + z = 180
Okay, these are our three equations.
Substitute 6*x for y in the last two equations to get:
z = 6*x + x^2 - 18
and
z = 180 - x - 6*x
Setting these two equal, we get:
6*x + x^2 - 18 = 180 - x - 6*x
This is a quadratic equation. You can solve it to find that x is 9.
Since y = 6*x, that means y = 54.
180 - 9 - 54 = 117
The three sides are 9, 54, and 117.
You can check these values with the original information in the problem to see that they work.
~ Mark

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10 = t/66 + 9
Solving for t means that we manipulate this equation so that t is all by itself on one side of the equals sign.
To start, let's get rid of the 9 on the right side by subtacting it. Of course, whatever arithmetic operation we do to one side of an equation we must do to both sides.
10 - 9 = t/66 + 9 - 9
1 = t/66
If we multiply t/66 by 66 we will get (66/66)*t. 66/66 is equal to 1, so that will isolate t all by itself. As usual, we must multiply BOTH sides by 66.
1*66 = (66/66)*t
This simplifies to:
66 = t
Let's check our answer by substituting 66 for t in the original equation to see if we get a true statement.
10 = t/66 + 9
10 = 66/66 + 9
10 = 1 + 9
10 = 10
This is a true statement, so our answer is correct.
~ Mark

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Hello There:
First of all, you should recognize the expression on the right side of the equals sign to be a quadratic expression. In other words, it's in the form of:
y = a*x^2 + b*x + c
Hopefully you know that the graph is going to be a parabola.
We have:
y = -x^2 + 16
Right away, we see that the parabola opens downward because the value of "a" is negative.
There are formulas for the x- and y-coordinates of the vertex point.
x-coordinate = -b/(2*a)
y-coordinate = c - b^2/(4*a)
Our equation shows that:
a = -1
b = 0
c = 16
Substitute these values into the formula above to find that (0, 16) is the vertex point.
The expression 16 - x^2 can be factored as a difference of squares.
(4 + x)*(4 - x)=0
So, the x-intercepts of the graph are (-4, 0) and (4, 0).
Pick some other values of x, and find the corresponding values of y to get a few more points to plot. Then draw the parabola using a smooth curve that goes through all of the points that you have plotted.
~ Mark

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Hello There:
I just responded to this question.
See:
Quadratic_Equations/20183
~ Mark

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Hello There:
If you think the number of tapes sold is 360, then you didn't understand the total problem. Both the number of tapes and CDs equals 360. The problem tells us that. So, if you choose 360 for the number of tapes, then that doesn't leave anything left over for the number of CDs.
Let x = the number of CDs sold
Then 4*x = the number of tapes sold
The given information tells us that:
x + 4*x = 360
5*x = 360
x = 360/5
x = 72
There were 72 CDs sold, so 4*72 is the number of tapes sold.
4*72 = 288
The correct answer is "a".
~ Mark

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w^2 + 3*w < 18
We recognize that subtracting 18 from both sides will lead to a quadratic expression on the left side of the inequality symbol. Let's do that.
w^2 + 3*w - 18 < 0
Next, we find the values of w that cause the left side to equal zero because these values are the boundary points between where the quadradic expression is larger or smaller than zero.
w^2 + 3*w - 18 = 0
The left side factors.
(w + 6)*(w - 3) = 0
So, we see that the values of -6 and 3 are the boundary points.
These two values divide the number line into three intervals. We pick a value within each interval to see if it leads to a correct statement when we substitute it into the original inequality given. (If one value works within an interval, then all of the values in that interval also work. This is why we only need to text one value in each interval.
For the interval (negative infinity to -6) we pick -10.
(-10)^2 + 3*(-10) < 18
100 - 30 < 18
70 < 18
This is a false statement, so the numbers in this interval are not solutions.
For the interval (-6, 3) we pick zero.
(0)^2 + 3*(0) < 18
0 < 18
This is true, so the values in this interval are solutions.
For the interval (3, positive infinity) we pick 10.
(10)^2 + 3*(10) < 18
100 + 30 < 18
130 < 18
This is false, so this interval is not a solution.
The solution is:
All values between -6 and 3.
~ Mark
P.S. This could also be done graphically by entering the expressions:
y1 = x^2 + 3*x
y2 = 18
on a graphing calculator, and then zooming in on the x-values where the bottom of the parabola lies beneath the line y = 18.