A line in the -plane passes through the points and . Which equation represents this line?

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SAT Linear Equations in Two Variables Question 139 | Free SAT Prep | Aniko

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Question 139·Easy·Linear Equations in Two Variables

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A line in the $xy$ -plane passes through the points $(0,1)$ and $(2,5)$ . Which equation represents this line?

For lines through two points, quickly compute the slope using $m = \frac{y_2 - y_1}{x_2 - x_1}$ to decide whether the line should have positive or negative slope and how steep it is, then plug one point into $y = mx + b$ to solve for $b$ . Once you know $m$ and $b$ , scan the answer choices for the one with that slope and intercept, eliminating any with the wrong sign for the slope or the wrong $y$ -intercept; this is faster and less error-prone than trying to test every option point-by-point.

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Think about slope

Use the two given points $(0,1)$ and $(2,5)$ to calculate the slope with $m = \frac{y_2 - y_1}{x_2 - x_1}$ .

Use slope-intercept form

Once you know the slope, write the equation in the form $y = mx + b$ and keep $m$ as the slope you found.

Use one of the points to find b

Substitute the coordinates of either $(0,1)$ or $(2,5)$ into $y = mx + b$ to solve for $b$ , then look for the choice with that $m$ and $b$ .

Desmos Guide

Plot the given points

Create a table in Desmos and enter the two points: one row with $x=0$ , $y=1$ and another with $x=2$ , $y=5$ . You should see these points appear on the graph.

Graph each answer choice

On separate lines in Desmos, type each option exactly as given: y = 2x - 1, y = -2x + 1, y = -2x - 1, and y = 2x + 1. Desmos will draw four lines.

Compare lines with points

Look at which of the four lines passes through both plotted points $(0,1)$ and $(2,5)$ . The equation of that line is the correct answer choice.

Step-by-step Explanation

Find the slope of the line

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

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

Here, use $(x_1,y_1) = (0,1)$ and $(x_2,y_2) = (2,5)$ :

m = \frac{5 - 1}{2 - 0} = \frac{4}{2} = 2

So the slope of the line is $2$ .

Use slope-intercept form and find the y-intercept

The slope-intercept form of a line is

y = mx + b

where $m$ is the slope and $b$ is the $y$ -intercept.

You already found $m = 2$ , so the equation looks like

y = 2x + b

Now use the fact that the line passes through $(0,1)$ . Substitute $x = 0$ and $y = 1$ into the equation:

1 = 2(0) + b \Rightarrow 1 = b

So the $y$ -intercept $b$ is $1$ .

Write the equation and match it to a choice

Now plug $m = 2$ and $b = 1$ into slope-intercept form:

y = 2x + 1

This is the equation of the line through $(0,1)$ and $(2,5)$ , which corresponds to choice D.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Step-by-step Explanation