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ACT Math : How to find the equation of a line

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Example Question #143 : Coordinate Geometry

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

Possible Answers:

–3x + 2y = 3

3x + 5y = 2

–2x + 2y = 3

2x – 3y = 5

4x – 5y = 4

Correct answer:

–3x + 2y = 3

Explanation:

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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Example Question #144 : Coordinate Geometry

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

Possible Answers:

$\dpi{100} \small y=-2x+3$

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

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

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

$\dpi{100} \small y=2x-3$

Correct answer:

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

Explanation:

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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Example Question #61 : Coordinate Geometry

A line is defined by the following equation:

7x+28y=84

What is the slope of that line?

Possible Answers:

$-\frac{1}{4}$

$\frac{1}{4}$

Correct answer:

$-\frac{1}{4}$

Explanation:

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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Example Question #1 : Lines

If the coordinates (3, 14) and (–5, 15) are on the same line, what is the equation of the line?

Possible Answers:

$y=-\frac{1}{8}x+13.625$

y=-8x+38

$y=\frac{1}{8}x+14.375$

$y=-\frac{1}{8}x+14.375$

y=-8x-38

Correct answer:

$y=-\frac{1}{8}x+14.375$

Explanation:

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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Example Question #1 : Coordinate Geometry

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

Possible Answers:

$\dpi{100} \small y=x+7$

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

$\dpi{100} \small y=2x+4$

$\dpi{100} \small y=2x-4$

$\dpi{100} \small y=3x+2$

Correct answer:

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

Explanation:

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 y=(-1)x+b . Use one of the give points in this equation to solve for the y-intercept. We will use $\dpi{100} \small (2,6)$ .

6=(-1)(2)+b

6=-2+b

8=b

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

y=(-1)x+8

y=-x+8

This is our final answer.

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Example Question #1 : How To Find The Equation Of A Line

Which of the following equations does NOT represent a line?

Possible Answers:

5y=10

x+y=10

x-y=10

x^2+y=10

x=10

Correct answer:

x^2+y=10

Explanation:

The answer is x^2+y=10 .

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

x^2+y=10 represents a parabola, not a line. Lines will never contain an x^2 term.

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Example Question #2 : Lines

Let y = 3x – 6.

At what point does the line above intersect the following:

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

Possible Answers:

(–3,–3)

(–5,6)

They do not intersect

(0,–1)

They intersect at all points

Correct answer:

They intersect at all points

Explanation:

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

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Example Question #1 : How To Find The Equation Of A Line

Which of the following is the equation of a line between the points (0,3) and (4,15) ?

Possible Answers:

4x + 3y = 4

y+2x=12

y-3x=3

2x - 3y = 3

3x - 2y = 13

Correct answer:

y-3x=3

Explanation:

Since you have y-intercept, this is very easy. You merely need to find the slope. Then you can use the form y=mx+b to find one version of the line.

The slope is:

$\frac{rise}{run} = \frac{y_1-y_0}{x_1-x_0}$

Thus, for the points (0,3) and (4,15) , it is:

$\frac{15-3}{4-0}= \frac{12}{4}=3$

Thus, one form of our line is:

y=3x + 3

If you move the to the left side, you get:

y-3x=3 , which is one of your options.

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Example Question #2 : How To Find The Equation Of A Line

What is an equation of the line going through points (4,8) and (6,10) ?

Possible Answers:

6x + 3y = 12

2x + 3y = 4

x-y=4

4x - 2y = 10

4y-4x=16

Correct answer:

4y-4x=16

Explanation:

If you have two points, you can always use the point-slope form of a line to find your equation. Recall that this is:

y-y_0=m(x - x_0)

You first need to find the slope, though. Recall that this is:

$\frac{rise}{run} = \frac{y_1-y_0}{x_1-x_0}$

For the points (4,8) and (6,10) , it is:

$\frac{10-8}{6-4}=\frac{2}{2} = 1$

Thus, you can write the equation using either point:

y-8=1(x-4)

y-x=4

Now, notice that one of the options is:

4y-4x=16

This is merely a multiple of the equation we found, so it is fine!

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Example Question #1 : How To Find The Equation Of A Line

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

Act_math_160_04

Possible Answers:

y=x-4

$y=\frac{10}{3}x-4$

y=-5x-4

y=-x

Correct answer:

$y=\frac{10}{3}x-4$

Explanation:

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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ACT Math : How to find the equation of a line

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #143 : Coordinate Geometry

Example Question #144 : Coordinate Geometry

Example Question #61 : Coordinate Geometry

Example Question #1 : Lines

Example Question #1 : Coordinate Geometry

Example Question #1 : How To Find The Equation Of A Line

Example Question #2 : Lines

Example Question #1 : How To Find The Equation Of A Line

Example Question #2 : How To Find The Equation Of A Line

Example Question #1 : How To Find The Equation Of A Line

All ACT Math Resources

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