| x | y | |---|---| | 0 | −2 | | 2 | 4 | | 4 | 10 | | 6 | 16 | The table above lists several pairs of -values and their corresponding -values for a linear relationship. Which equation models this relationship?

Solution available with step-by-step explanation

SAT Linear Equations in Two Variables Question 112 | Free SAT Prep | Aniko

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Question 112·Easy·Linear Equations in Two Variables

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x	y
0	−2
2	4
4	10
6	16

The table above lists several pairs of $x$ -values and their corresponding $y$ -values for a linear relationship. Which equation models this relationship?

For table-to-equation linear questions, first confirm the relationship is linear by checking that the change in $y$ is constant for equal changes in $x$ . Compute the slope quickly using $m = \dfrac{\Delta y}{\Delta x}$ from any two convenient points, then get the $y$ -intercept from the row where $x=0$ (or by plugging a point into $y = mx + b$ and solving for $b$ ). Finally, match the slope and intercept you found to the answer choice in $y = mx + b$ form, and always double-check by mentally plugging in one point from the table.

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Look at how y changes as x increases

Compare the $y$ -values in the table each time $x$ increases by $2$ . Is the change in $y$ always the same? What does that tell you about the slope?

Use two points to compute the slope

Pick any two points from the table, for example $(0,-2)$ and $(2,4)$ , and calculate $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ . Keep that slope in mind when you look at the answer choices.

Find the y-intercept from the table

In the form $y = mx + b$ , $b$ is the $y$ -value when $x=0$ . Which row in the table tells you that, and what is the intercept?

Match to the choices

Now that you know the slope and the $y$ -intercept, look at each equation and see which one has both the same slope and the same intercept.

Desmos Guide

Enter the table of points

Create a table in Desmos and enter the four points from the problem: $(0,-2)$ , $(2,4)$ , $(4,10)$ , and $(6,16)$ . You should see these points plotted on the coordinate plane.

Graph each answer choice

Type each option as a separate equation line in Desmos: y = 2x + 4, y = 4x - 2, y = x - 2, and y = 3x - 2. Check, one by one, whether the line passes exactly through all four plotted points from the table.

Identify the matching line

The correct equation is the one whose graph goes through every point in the table without missing any. Use the graph to see which line lines up perfectly with all four points.

Step-by-step Explanation

Find the rate of change (slope) from the table

Look at how $y$ changes when $x$ increases by the same amount.

From $x=0$ to $x=2$ :

$y$ goes from $-2$ to $4$ , so the change in $y$ is $4 - (-2) = 6$ .
The change in $x$ is $2 - 0 = 2$ .

So the slope is

m = \dfrac{\Delta y}{\Delta x} = \dfrac{6}{2} = 3

Check it stays the same:

From $x=2$ to $x=4$ : $y$ goes $4 \to 10$ , change is $6$ again, over $2$ in $x$ .
From $x=4$ to $x=6$ : $y$ goes $10 \to 16$ , again change $6$ over $2$ in $x$ .

So the relationship is linear with slope $3$ .

Identify the y-intercept from the table

In a linear equation $y = mx + b$ , $b$ is the $y$ -intercept: the $y$ -value when $x=0$ .

From the table, when $x=0$ , $y=-2$ . That means the $y$ -intercept $b$ is $-2$ .

So we now know:

Slope $m = 3$
Intercept $b = -2$

Match the slope and intercept to the correct equation

We want an equation of the form $y = mx + b$ with slope $m=3$ and $y$ -intercept $b=-2$ .

Check each option:

A) $y = 2x + 4$ has slope $2$ and intercept $4$ .
B) $y = 4x - 2$ has slope $4$ and intercept $-2$ .
C) $y = x - 2$ has slope $1$ and intercept $-2$ .
D) $y = 3x - 2$ has slope $3$ and intercept $-2$ .

Only choice D has both the correct slope and intercept.

Answer: D) $y = 3x - 2$ .

Strategy

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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