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PSAT 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 #21 : Geometry

Axes

Refer to the above red line. What is its equation in standard form?

Possible Answers:

3x + y = 18

x + 3y = 6

x -3y = - 6

x + 3y = - 6

3x - y = -18

Correct answer:

3x - y = -18

Explanation:

First, we need to find the slope of the above line.

The slope of a line. given two points $(x_{1}, y_{1}), (x_{2}, y_{2})$ can be calculated using the slope formula

$m = \frac{y_{2}-y_{1}}{x_{2}-x _{1}}$

Set $x_{1}=-6, y_{1}=x_{2}= 0, y_{2}=18$ :

$m = \frac{18-0}{0-(-6)} = \frac{18}{6} = 3$

Second, we note that the -intercept is the point (0,18) .

Therefore, in the slope-intercept form of a line, we can set m = 3 and b = 18 :

y = mx+b

y = 3x + 18

Since we are looking for standard form - that is, Ax+By = C - we do the following:

y - y - 18 = 3x + 18 - y - 18

- 18 = 3x - y

3x - y = -18

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Example Question #21 : Coordinate Geometry

Axes

Refer to the above red line. What is its equation in slope-intercept form?

Possible Answers:

y = 3x -6

y = 3x + 18

$y = \frac{1}{3}x -6$

$y = \frac{1}{3}x + 18$

$y = \frac{1}{3}x +6$

Correct answer:

y = 3x + 18

Explanation:

First, we need to find the slope of the above line.

The slope of a line. given two points $(x_{1}, y_{1}), (x_{2}, y_{2})$ can be calculated using the slope formula

$m = \frac{y_{2}-y_{1}}{x_{2}-x _{1}}$

Set $x_{1}=-6, y_{1}=x_{2}= 0, y_{2}=18$ :

$m = \frac{18-0}{0-(-6)} = \frac{18}{6} = 3$

Second, we note that the -intercept is the point (0,18) .

Therefore, in the slope-intercept form of a line, we can set m = 3 and b = 18 :

y = mx+b

y = 3x + 18

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Example Question #21 : Geometry

Axes

Refer to the above red line. A line is drawn perpendicular to that line, and with the same -intercept. What is the equation of that line in slope-intercept form?

Possible Answers:

$y = \frac{1}{3}x+6$

$y = -\frac{1}{3}x-2$

$y = -\frac{1}{3}x+6$

$y = -\frac{1}{3}x-6$

$y = \frac{1}{3}x-2$

Correct answer:

$y = -\frac{1}{3}x-2$

Explanation:

First, we need to find the slope of the above line.

The slope of a line. given two points $(x_{1}, y_{1}), (x_{2}, y_{2})$ can be calculated using the slope formula

$m = \frac{y_{2}-y_{1}}{x_{2}-x _{1}}$

Set $x_{1}=-6, y_{1}=x_{2}= 0, y_{2}=18$ :

$m = \frac{18-0}{0-(-6)} = \frac{18}{6} = 3$

The slope of a line perpendicular to it has as its slope the opposite of the reciprocal of 3, which is $m = -\frac{1}{3}$ .

Since we want the line to have the same -intercept as the above line, which is the point (-6,0) , we can use the slope-intercept form to help us. We set

$m = -\frac{1}{3},x= -6, y=0$ , and solve for in:

y = mx+b

$0 = -\frac{1}{3}(-6)+b$

0 =2 +b

b = -2

Substitute for and in the slope-intercept form, and the equation is

$y = -\frac{1}{3}x-2$

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Example Question #61 : Other Lines

Find the equation of the line shown in the graph below:

Sat_math_164_05

Possible Answers:

y = x/2 + 4

y = -1/2x + 4

y = -1/2x - 4

y = 2x + 4

Correct answer:

y = x/2 + 4

Explanation:

Based on the graph the y-intercept is 4. So we can eliminate choice y = x/2 - 4.

The graph is rising to the right which means our slope is positive, so we can eliminate choice y = -1/2x + 4.

Based on the line, if we start at (0,4) and go up 1 then 2 to the right we will be back on the line, meaning we have a slope of (1/2).

Using the slope intercept formula we can plug in y= (1/2)x + 4.

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PSAT Math : How to find the equation of a line

Study concepts, example questions & explanations for PSAT Math

All PSAT 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 #21 : Geometry

Example Question #21 : Coordinate Geometry

Example Question #21 : Geometry

Example Question #61 : Other Lines

All PSAT Math Resources

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