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Factor:
Try this:

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Provide the missing numbers:
-1, ?, -3 1/2, ?, -6, -7 1/4, ?, ?,
First, you'll notice that the numbers are getting smaller (more negative) so it is possible that subtraction involved.
You can test this on the two adjacent numbers in the set...-6, and -7 1/4
You can see that if you subtract -7 1/4 from -6 you'll get a difference of
1 1/4.
So, if you subtract 1 1/4 from the first number (-1) you'll get -2 1/4
Now subtract 1 1/4 from -2 1/4 and you'll get -3 1/2...so it seem to work with a common difference of 1 1/4
The missing numbers are:
-2 1/4, -4 3/4, -8 1/2, -9 3/4.

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Sorry, but no cigar!
You do have the first term corect ( but the y-intercept (b) is not right.
Let's start with what you have done.
To find the value of b, you need to substitute the x- and y-coordinates of the given point (2, 1)
Simplify this.
Subtract 5 from both sides of the equation.
Now you can write the equation:

Here are the two lines on a graph:
Red line.
Green line.

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Simplify:
Rewrite this as:
Now you can take the square root of 4 (that's 2) and put it outside the radical.
Simplifying this, you get:

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Let's write the required equation of the line in the "slope-intercept" form: Where: m is the slope of the line and b is the y-intercept of the line.
Remember that parallel lines have the same slope.
So, your new line is parallel the the line whose equation is:
Comparing this with the slope-intercept form:
you can see that the slope of this line is and, because the new line is parallel to this, it too will have a slope of .
So, step 1, for the new line, you can write:
Now all you need to do is to find the value of b, the y-intercept of the new line.
You can do this by substituting the x- and y-coordinates of the given point (2, 1) into the equation and solve for b.
Simplify this.
Subtract 5 from both sides of the equation.
You can write the final equation now that you have the slope and the y-intercept :
...and that's it!

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Ok. give your brain a rest and let's see what can be done with this:
Solve:
for b. First, subtract from both sides.
Now combine the two fractions on right side.
Next, multiply both sides by ac
Then multiply both sides by b
Finally, divide both sides by (3a-c)

So,

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I assume that you meant to write..."Each pen will be three times as long as it is wide,..."
If we let the width be W, then the length of each pen will be: L = 3W.
From the problem description, the two adjacent pens will have three long sides and four short sides.
Each long side will be 3W in length and there are three of these,
so we have 3(3W) = 9W. This is the total of the lengths.
The short sides (Widths) wll each be W in length and there are four of these, so we have 4(W) = 4W This the total of the widths.
Adding up all of these, we get:
9W + 4W = 13W and this is equal to the total length of the fence (perimeter) of 65 feet.
13W = 65 Divide both sides by 13.
W = 5 feet.
This is the length of each width, but there are four of these, so we have the total for the widths is: 4(5) = 20 feet.
The length, L = 3W = 3(5) = 15 feet
The three lengths is then: 3(15) = 45 feet.
Each pen measures: (L X W) = 15 ft. by 5 ft.
Check:
Adding up the the total widths and the total lengths, we get: 20 ft + 45 ft = 65 ft.

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Michael paid a total of $95.74 when he returned the truck. So, if we first deduct (subtract) the rental fee of $20.99 to see what he actually paid for mileage. Then we can divide the remainder by $0.25 to see how many miles he drove.
(Total cost - rental fee)/$0.25 = number of miles driven.
($95.74 - $20.99}/$0.25 = ($74.75)/$0.25 = 299
So Michael paid for 299 miles

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Find the values of m and b given the solution to the system is (-3, 4):
5x + 7y = b
mx +y = 22
Substitute the x- and y-coordinates from the given point (-3, 4) into each equation and solve for the b and m.
5(-3) + 7(4) = b Simplify.
-15 + 28 = b
b = 13
mx + y = 22
m(-3) + 4 = 22 Subtract 4 from both sides of the equation.
-3m = 18 Divide both sides by -3.
m = -6
You can verify these solutions by substituting b = 13 and m = -6 into the appropriate equation and doing the math.
Try this yourself.

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Solve:
Add 9 to both sides of the equation.
Square both sides.
Subtract 3 from both sides.
Divide both sides by 3.
Check:

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Graph:
The red line.
The green line.

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The first step, of course, is to find the volume of the circular (really cylindrical) swimming pool.
Where: r (radius) is 6 ft. and h (depth) is 4 ft. Making these substitutions:


cubic feet.
The next step is to discover how many cubic feet are occupied by 1 U.S. gallon of water.
1 U.S. gallon = 0.1336816 cubic feet.
So, how many of these will fit into 452.16 cubic feet? This sounds like a division problem, so we'll divide 452.16 cubic feet by 0.0336816 cubic feet /gallon.
Gallons (Aproximately)

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Let's answer all three questions for the problem here.
Since the initial upward velocity is given as 32 ft/sec, you can write the function of height,s, as a function of time, t. If you were to graph this function, you would get a parabola that opens downward. We know that because the coefficient of the t^2 term is negative.
For question #2, the time at which maximum height is attained by the baseball can be found by locating the vertex of the parabola then finding the value of t that corresponds to this location.
The t-coordinate of the vertex is given by: .
Where do the a and b come from?
They come from comparing your equation with the general standard form of the quadratic equation: but in your equation of course, the independent variable is t rather than x and a = -16 and b = 32.
So, let's find the the value of t at which the vertex is located.
In your equation, a = -16 and b = 32. Substituting these values into the appropriate places, we get:
Simplifying this:
second.
The baseball attains its maximum height in 1 second.
For question #1, the maximum height attained by the baseball can be found by substituting t = 1 in the original function for the height: and solveing for the height, s.
Simplifying, we get:

feet.
The maximum height attained by the baseball is 16 feet.
Question #3 (Posted separately) At what time does the baseball return to the ground? Well, that's really asking at what time is the height of the base = 0 feet. There will, of course, be two anwers to this question. The height will be zero at the start of the action and then it will be zero when the baseball returns to earth, so only the second solution is of interest to us. To find this time, we'll set the function for the height equal to zero and solve for the time, t.
Factor a t.
Apply the zero product principle.
and/or
Well, t = 0 is the first solution whch we can ignore.
Subtract 32 from both sides of the equation.
Divide both sides by -16.
seconds.
The baseball returns to the ground at 2 seconds.

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Since you are given the lengths of only the two legs, you need to find the length of the hypotenuse of this right triangle before you can find the perimeter. Use the Pythagorean theorem to find the length of the hypotenuse, c.

Substitute a = 5 and b = 9



cm. (Approximately} The perimeter is:


cm (Approximately)

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Multiply:
Use FOIL to multiply these two factors.
Simplify this.

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I know this is algebra and it seems like everything in algebra requires an equation of some sort. But sometimes, a little common sense will get the job done with a lot less confusion.
Try this.
The definition of slope is "rise over run".
In your problem, you could say Tom "rose" (in reverse) 1800 feet while he "ran" (horizontally) 3.25 miles.
So the slope here is Rise/Run = 1800 ft./3.25 miles.
But you'll need to change the units so that they are the same, either all in feet or all in miles. All in feet seems to be the better approach, so how many feet are there in 3.25 miles?
Well, 1 mile = 5280 ft. Multiply both sides by 3.25
3.25 mies = 3.25(5280) ft.
3.25 miles = 17160 ft. Now we can calculate the slope:
But you need to round to the nearest hundreth.
The slope is 0.10

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Write the equation in slope-intercept form:

You are given the slope, m = 4, so you can write:
Now you need to find the value of b, the y-intercept. This is easily done because you were also given a point (2, 8) through which the line passes.
So you substitute the (x, y) of the given popint (2, 8) into the equation and solve for b.
Simplify.
Subtract 8 from both sides of the equation.
So b = 0. Now you can write the final equation of the line whose slope is 4 and which passes through the point (2, 8).

Just out of curiosity, let's see what that looks like on a graph:

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The volume of a right circular cylinder is given by:
Let's first solve this for the height, h, by dividing both sides by .
Now you can substitute , and
You can cancel the and
Simplify.
cm.

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It seems to me that you are into "factoring" trinomials, which is basically a process of converting trinomials into binomial factors. Of course, not all trinomials can be factored, but "factoring trinomials" constitutes an entire lesson for those that can be factored.
Here's an elementary example:
Factor the following expression:

The factors will be two "binomials" of the following form:
You'll need to find the values of a and b. These two binomial factors, when multiplied together will result in the original trinomial expression.
Comparing the binomial factors with the trinomial, you can write this:
so this means that:
a+b = 3 and a*b = 2
What are the possibilities?
a = 1 and b = 2 or
a = 2 and b = 1
Let's try these in the binomial forms and see what we get.
Multiply these factors using the FOIL method.
Simplifying this, we get:
The original trinomial!

You would get exactly the same result if you used:
a = 2 and b = 1

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Remember the basic definition of slope..."Rise over Run"
In your problem, the aqueduct rises 3 meters (Rise = 3) for every kilometer (1,000 meters) in length (Run = 1,000 meters).
So the slope is Rise/Run = 3/1000 = 0.003
Does this help?

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Presumably you want to solve for M?
Divide both sides of the inequality by 25.

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I'm not certain that the following constitutes a "proof" in the formal sense of the word, but it certainly provides a demonstration.
If you accept the principle that "An angle inscribed in a semi-circle is a right angle", then read the following:
The line segment LM is the diameter of a circle.
The mid-point of LM is therefore the center of this circle and PL and PM are both radii of this circle and they, of course, are equal.
Point K is a point on the circumference of the circle, not coincident with either L or M.
The segment PK is also a radius of the circle, so that:
PL = PK = PM
Segments KL and KM form the other two sides of a triangle.
The angle LKM is inscribed in a semi-circle and is, therefore, a right angle. A right angle = 90 degrees.

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Express x in terms of a and b.
Add 5.6 to both sides of the equation.
Now subtract bx from both sides.
Factor out the x.
Finally, divide both sides by (a-b)

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Find the two solutins:
First, factor out an x.
Apply the zero product principle.
x = 0 and/or ax+8a = 0
x = 0 is one of the two solutions.
Subtract 8a from both sides.
Divide both sides by a.

x = -8 is the other solution.

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Solving for y simply means that you must get the y on one side of the equals sign by itself using the standard rules of algebra.
1) 5x-y = 10 Here, you would first add y to both sides of the equation.
5x = 10+y Then you subtract 10 from both sides.
5x-10 = y Many teachers like to see the dependent variable (y in this case) on the left side, so rewrite this as:
y = 5x-10...and there you have it.
2) x+4y = -8 In this one, you would subtract x from both sides.
4y = -x-8 then you divide both sides by 4.
y = -(1/4)x - 2
Do you see the idea here? You want to rewrite each equation so that it's in the form...y = something. Let's do one more:
0.1x + 0.5y = 3.5 Here, you need to get rid of the decimals in a way that leaves you with integral (whole number) numbers of y. You can do this by multiplying everything by 2.
(2)0.1x + (2)0.5y = (2)3.5 Simplify this.
0.2x + y = 7 Now subtract 0.2x from both sides.
y = -0.2x + 7
Ok now try the others on your own.
If you are still having trouble with it, repost.

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It seems as though you might have missed part of the problem statement.
Do you have to find the height of the rock 5 seconds after it is dropped from the North Rim at a height of 5700 feet? Using the given function:
Substitute t = 5 and solve for h.


feet.
The height of the rock would be 5300 feet five seconds after it was dropped from a height of 5700 feet.

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Well, it looks like you haven't lost your touch.
Your initial step was quite correct. I am assuming that the task is to simplify the given expression.
You correctly factored the numerator of the first fraction.
and you also correctly cancelled the (x+8)'s
At this point, you can cancel 5x's in the denominator and from the in the numerator.
Finally, you can multiply these two factors.

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Step 1.
Rewrite your equation in the form of a quadratic function.
This is function for the height, h, of an object propelled upwards as a function of time, t. Here, r is given as 180 ft/sec. This fits the situation of Mark's rocket. You are being asked to find at what time, t, will the height, h, be 464 feet. Because your equation is quadratic, you can expect to get two solutions.
Step 2. Set h(t) = 464 in the quadratic equation above and solve for t.
Put this into standard form by subtracting 464 from both sides.

Step 3. Solve the quadratic equation for t by the most convenient method. You can simplify this equation a bit by dividing through by -4 to get:
Now you can factor this.
Apply the zero product principle:
and/or
If then t = 4
If then or
The two solution are:
The height of the rocket will reach 464 feet in 4 seconds, ascending.
The height of the rocket will be at 464 feet again in 7.25 seconds descending.

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Solve for x:
The idea is to get the x onto one side of the equals sign. Start by subtracting 3x fom both sides of the equation.
Now add 16 to both sides.
Finally, divide both sides by 2.

Check: Set x = 8 in the original equation.
Simplify.

It checks!

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If n is the number, it would look like:

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Let the sides of the square = s
The perimeter would be:

The area of the square would be:
Take the square root of both sides.
Subsitute this s into the equation for the perimeter.