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SAT Math : How to find the equation of a line

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Example Question #137 : Coordinate Geometry

$\small \textup{Which is the equation of the line that passes through the points}$

$\small (1,2)\,\textup{and}\,(3.8)?$

Possible Answers:

$\small y=\frac{x}{3}+1$

$\small y = x+5$

$\small y=3x-1$

$\small y=3x+2$

$\small y=-3x +2$

Correct answer:

$\small y=3x-1$

Explanation:

Equation of line: $\small y = mx+b$ , $\small m$ = slope, $\small b$ = -intercept

Step 1) Find slope ( $\small m$ ): rise/run $\small \frac{8-2}{3-1}=\frac{6}{2}=3$

Step 2) Find -intercept ( $\small b$ ): $\small \frac{2-b}{1-0}=3$

$\small 2-b = 3$

$\small 2 = 3 + b$

$\small -1 =b$

$\small y=3x-1$

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Example Question #138 : Coordinate Geometry

Whast line goes through the points (1,3) and (7,5) ?

Possible Answers:

-x+3y=8

4x+3y=7

x-2y=6

x+y=4

3x-2y=9

Correct answer:

-x+3y=8

Explanation:

Let $P_{1}=(1,3)$ and $P_{2}=(7,5)$

The slope is geven by: $m = (y_{2} - y_{1}) \div (x_{2} - x_{1})$ so

$m=\frac{1}{3}$

Then we use the slope-intercept form of an equation; y=mx+b so

$b=\frac{8}{3}$

And we convert

$y=\frac{1}{3}x+\frac{8}{3}$

to standard form.

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Example Question #11 : How To Find The Equation Of A Line

What is the equation of the line that passes through the points (4,7) and (8,10)?

Possible Answers:

3x-4y=16

-3x-4y=16

3x+4y=-16

3x+4y=16

3x-4y=-16

Correct answer:

3x-4y=-16

Explanation:

In order to find the equation of the line, we will first need to find the slope between the two points through which it passes. The slope, , of a line that passes through the points (x_1,y_1) and (x_2,y_2) is given by the formula below:

$\text{slope}=m=\frac{y_2-y_1}{x_2-x_1}$

We are given our two points, (4,7) and (8,10), allowing us to calculate the slope.

$m=\frac{10-7}{8-4}=\frac{3}{4}$

Next, we can use point slope form to find the equation of a line with this slope that passes through one of the given points. We will use (4,7).

y-y_1=m(x-x_1)

$y-7=\frac{3}{4}(x-4)$

Multiply both sides by four to eliminate the fraction, and simplify by distribution.

$4(y-7)=4(\frac{3}{4})(x-4)$

4y-28=3(x-4)

4y-28=3x-12

Subtract from both sides and add twelve to both sides.

-28=3x-4y-12

-16=3x-4y

This gives our final answer: 3x-4y=-16

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Example Question #140 : Coordinate Geometry

Which line contains the following ordered pairs:

$\left ( -2,4 \right )$ and $\left ( 6,2 \right )$

Possible Answers:

$\small y=\frac{1}{4}x+\frac{7}{2}$

$\small y=-x+14$

$\small y=x+14$

$\small y=-\frac{1}{4}x+\frac{7}{2}$

Correct answer:

$\small y=-\frac{1}{4}x+\frac{7}{2}$

Explanation:

First, solve for slope.

$\small m=\frac{\Delta y}{\Delta x}=\frac{2-4}{6-(-2)}=\frac{-2}{8}=-\frac{1}{4}$

Then, substitute one of the points into the equation y=mx+b.

$\small 2=(-\frac{1}{4})(6)+b$

$\small 2=(-\frac{3}{2})+b$

$\small b=2+\frac{3}{2}=\frac{7}{2}$

This leaves us with the equation $\small y=-\frac{1}{4}+\frac{7}{2}$

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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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Example Question #241 : Geometry

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

Possible Answers:

y = 1/5x + 3

y = 2/3x –6

y = –3/2 – 3

y = 2/3 + 5

y = –3/2x + 6

Correct answer:

y = –3/2x + 6

Explanation:

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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Example Question #1 : Coordinate Geometry

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

Possible Answers:

–2x + 2y = 3

4x – 5y = 4

–3x + 2y = 3

2x – 3y = 5

3x + 5y = 2

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 #1 : 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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SAT Math : How to find the equation of a line

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #137 : Coordinate Geometry

Example Question #138 : Coordinate Geometry

Example Question #11 : How To Find The Equation Of A Line

Example Question #140 : Coordinate Geometry

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

Example Question #241 : Geometry

Example Question #1 : Coordinate Geometry

Example Question #144 : Coordinate Geometry

Example Question #1 : Coordinate Geometry

Example Question #1 : Lines

All SAT Math Resources

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