How to find the equation of a line - GRE Quantitative Reasoning

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Question

If the coordinates (3, 14) and (_–_5, 15) are on the same line, what is the equation of the line?

Answer

First solve for the slope of the line, m using y=mx+b

m = (y2 – y1) / (x2 – x1)

= (15 – 14) / (_–_5 _–_3)

= (1 )/( _–_8)

=_–_1/8

y = –(1/8)x + b

Now, choose one of the coordinates and solve for b:

14 = –(1/8)3 + b

14 = _–_3/8 + b

b = 14 + (3/8)

b = 14.375

y = –(1/8)x + 14.375

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Question

What is the equation of the line passing through (–1,5) and the upper-right corner of a square with a center at the origin and a perimeter of 22?

Answer

If the square has a perimeter of 22, each side is 22/4 or 5.5. This means that the upper-right corner is (2.75, 2.75)—remember that each side will be "split in half" by the x and y axes.

Using the two points we have, we can ascertain our line's equation by using the point-slope formula. Let us first get our slope:

m = rise/run = (2.75 – 5)/(2.75 + 1) = –2.25/3.75 = –(9/4)/(15/4) = –9/15 = –3/5.

The point-slope form is: y – y0 = m(x – x0). Based on our data this is: y – 5 = (–3/5)(x + 1); Simplifying, we get: y = (–3/5)x – (3/5) + 5; y = (–3/5)x + 22/5

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Question

Which line passes through the points (0, 6) and (4, 0)?

Answer

P1 (0, 6) and P2 (4, 0)

First, calculate the slope: m = rise ÷ run = (y2 – y1)/(x2 – x1), so m = –3/2

Second, plug the slope and one point into the slope-intercept formula:

y = mx + b, so 0 = –3/2(4) + b and b = 6

Thus, y = –3/2x + 6

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Question

What line goes through the points (1, 3) and (3, 6)?

Answer

If P1(1, 3) and P2(3, 6), then calculate the slope by m = rise/run = (y2 – y1)/(x2 – x1) = 3/2

Use the slope and one point to calculate the intercept using y = mx + b

Then convert the slope-intercept form into standard form.

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Question

Let y = 3_x_ – 6.

At what point does the line above intersect the following:

$2x =\frac{2}{3}y+4$

Answer

If we rearrange the second equation it is the same as the first equation. They are the same line.

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Question

Given the graph of the line below, find the equation of the line.

Answer

To solve this question, you could use two points such as (1.2,0) and (0,-4) to calculate the slope which is 10/3 and then read the y-intercept off the graph, which is -4.

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Question

What is the equation of the straight line passing through (–2, 5) with an x-intercept of 3?

Answer

First you must figure out what point has an x-intercept of 3. This means the line crosses the x-axis at 3 and has no rise or fall on the y-axis which is equivalent to (3, 0). Now you use the formula (y2 – y1)/(x2 – x1) to determine the slope of the line which is (5 – 0)/(–2 – 3) or –1. Now substitute a point known on the line (such as (–2, 5) or (3, 0)) to determine the y-intercept of the equation y = –x + b. b = 3 so the entire equation is y = –x + 3.

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Question

A line is defined by the following equation:

What is the slope of that line?

Answer

The equation of a line is

y=mx + b where m is the slope

Rearrange the equation to match this:

7x + 28y = 84

28y = -7x + 84

y = -(7/28)x + 84/28

y = -(1/4)x + 3

m = -1/4

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Question

What is the equation of a line that passes through coordinates $\dpi{100} \small (2,6)$ and $\dpi{100} \small (3,5)$ ?

Answer

Our first step will be to determing the slope of the line that connects the given points.

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{6-5}{2-3}=\frac{1}{-1}=-1$

Our slope will be . Using slope-intercept form, our equation will be . Use one of the give points in this equation to solve for the y-intercept. We will use $\dpi{100} \small (2,6)$ .

Now that we know the y-intercept, we can plug it back into the slope-intercept formula with the slope that we found earlier.

This is our final answer.

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Question

What is the slope-intercept form of $\dpi{100} \small 8x-2y-12=0$ ?

Answer

The slope intercept form states that $\dpi{100} \small y=mx+b$ . In order to convert the equation to the slope intercept form, isolate $\dpi{100} \small y$ on the left side:

$\dpi{100} \small 8x-2y=12$

$\dpi{100} \small -2y=-8x+12$

$\dpi{100} \small y=4x-6$

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Question

Which of the following equations does NOT represent a line?

Answer

The answer is .

A line can only be represented in the form or , for appropriate constants , , and . A graph must have an equation that can be put into one of these forms to be a line.

represents a parabola, not a line. Lines will never contain an term.

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Question

What is the equation of a line passing through with a -intercept of ?

Answer

Based on the information that you have been provided, you can quickly find the slope of your line. Since the y-intercept is , you know that the line contains the point . Therefore, the slope of the line is found:

$\frac{-12-9}{4-0}=\frac{-21}{4}$

Based on this information, you can use the standard slope-intercept form to find your equation:

, where $m=\frac{-21}{4}$ and

$y=\frac{-21}{4}x+9$

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Question

What is the equation of a line passing through the points and ?

Answer

Based on the information provided, you can find the slope of this line easily. From that, you can use the point-slope form of the equation of a line to compute the line's full equation. The slope is merely:

$\frac{rise}{run}=\frac{17-(-11)}{-2-5}=\frac{17+11}{-7}=\frac{-28}{7}=-4$

Now, for a point and a slope , the point-slope form of a line is:

Let's use for our point

This gives us:

Now, distribute and solve for :

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Question

What is the equation of a line passing through the two points and ?

Answer

Based on the information provided, you can find the slope of this line easily. From that, you can use the point-slope form of the equation of a line to compute the line's full equation. The slope is merely:

$\frac{rise}{run}=\frac{11-(-9)}{41-4}=\frac{11+9}{37}=\frac{20}{37}$

Now, for a point and a slope , the point-slope form of a line is:

Let's use for our point

This gives us:

$y+9=\frac{20}{37}(x-4)$

Now, distribute and solve for :

$y+9=\frac{20}{37}x-\frac{80}{37}$

$y=\frac{20}{37}x-\frac{80}{37}-9$

$y=\frac{20}{37}x-\frac{80}{37}-\frac{333}{37}$

$y=\frac{20}{37}x-\frac{413}{37}$

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Question

If the coordinates (3, 14) and (_–_5, 15) are on the same line, what is the equation of the line?

Answer

First solve for the slope of the line, m using y=mx+b

m = (y2 – y1) / (x2 – x1)

= (15 – 14) / (_–_5 _–_3)

= (1 )/( _–_8)

=_–_1/8

y = –(1/8)x + b

Now, choose one of the coordinates and solve for b:

14 = –(1/8)3 + b

14 = _–_3/8 + b

b = 14 + (3/8)

b = 14.375

y = –(1/8)x + 14.375

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Question

What is the equation of the line passing through (–1,5) and the upper-right corner of a square with a center at the origin and a perimeter of 22?

Answer

If the square has a perimeter of 22, each side is 22/4 or 5.5. This means that the upper-right corner is (2.75, 2.75)—remember that each side will be "split in half" by the x and y axes.

Using the two points we have, we can ascertain our line's equation by using the point-slope formula. Let us first get our slope:

m = rise/run = (2.75 – 5)/(2.75 + 1) = –2.25/3.75 = –(9/4)/(15/4) = –9/15 = –3/5.

The point-slope form is: y – y0 = m(x – x0). Based on our data this is: y – 5 = (–3/5)(x + 1); Simplifying, we get: y = (–3/5)x – (3/5) + 5; y = (–3/5)x + 22/5

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Question

Which line passes through the points (0, 6) and (4, 0)?

Answer

P1 (0, 6) and P2 (4, 0)

First, calculate the slope: m = rise ÷ run = (y2 – y1)/(x2 – x1), so m = –3/2

Second, plug the slope and one point into the slope-intercept formula:

y = mx + b, so 0 = –3/2(4) + b and b = 6

Thus, y = –3/2x + 6

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Question

What line goes through the points (1, 3) and (3, 6)?

Answer

If P1(1, 3) and P2(3, 6), then calculate the slope by m = rise/run = (y2 – y1)/(x2 – x1) = 3/2

Use the slope and one point to calculate the intercept using y = mx + b

Then convert the slope-intercept form into standard form.

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Question

Let y = 3_x_ – 6.

At what point does the line above intersect the following:

$2x =\frac{2}{3}y+4$

Answer

If we rearrange the second equation it is the same as the first equation. They are the same line.

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Question

Given the graph of the line below, find the equation of the line.

Answer

To solve this question, you could use two points such as (1.2,0) and (0,-4) to calculate the slope which is 10/3 and then read the y-intercept off the graph, which is -4.

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