In the -plane, line passes through the points and . Which equation defines line ?

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SAT Linear Equations in Two Variables Question 41 | Free SAT Prep | Aniko

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Question 41·Medium·Linear Equations in Two Variables

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In the $xy$ -plane, line $w$ passes through the points $(-4,3)$ and $(2,-9)$ . Which equation defines line $w$ ?

For line-through-two-points questions, quickly compute the slope using $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ , then immediately eliminate any answer choices with a different slope (different coefficient of $x$ ). With the correct slope in hand, either write $y = mx + b$ and plug in one point to solve for $b$ , or plug a point directly into the remaining choices to see which one gives a true statement. This minimizes arithmetic and helps you avoid sign errors under time pressure.

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Start by finding the slope

Use the slope formula $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ with the two given points. Be careful with the subtraction and the negative signs.

Use slope-intercept form

Once you know the slope $m$ , write the equation as $y = mx + b$ . Then you only need to find the value of $b$ .

Plug a point into the equation

Take one of the given points and substitute its $x$ and $y$ values into $y = mx + b$ to solve for $b$ . Then compare your final equation with the answer choices.

Desmos Guide

Plot the given points

In Desmos, type (-4,3) and (2,-9) on separate lines so you can see the two points that line $w$ must pass through.

Graph each answer choice

On new lines in Desmos, enter each option exactly as written: y = 2x - 5, y = -2x + 5, y = 2x + 5, and y = -2x - 5. You will see four different lines on the graph.

See which line goes through both points

Look at the graphs and check which of the four lines passes exactly through both plotted points $(-4,3)$ and $(2,-9)$ . The equation of that line is the correct choice for line $w$ .

Step-by-step Explanation

Find the slope using the two points

Use the slope formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ :

m = \dfrac{y_2 - y_1}{x_2 - x_1}

Let $(-4,3)$ be $(x_1,y_1)$ and $(2,-9)$ be $(x_2,y_2)$ .

Then

m = \dfrac{-9 - 3}{2 - (-4)} = \dfrac{-12}{6} = -2

So the slope of line $w$ is $-2$ .

Write the general equation with the known slope

A line with slope $m$ can be written in slope-intercept form as $y = mx + b$ , where $b$ is the $y$ -intercept.

We found $m = -2$ , so line $w$ must have an equation of the form

y = -2x + b

Among the answer choices, only the ones with slope $-2$ (the coefficient of $x$ is $-2$ ) can be correct.

Use one point to find the y-intercept and match the choice

Substitute one of the points, say $(-4,3)$ , into $y = -2x + b$ to solve for $b$ :

3 = -2(-4) + b

3 = 8 + b

b = -5 $$ So the equation of line $w$ is $y = -2x - 5$, which corresponds to choice D.

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation