SAT Linear Equations in Two Variables Question 24 | Free SAT Prep

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Question 24·Hard·Linear Equations in Two Variables

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In the $xy$ -plane, a line passes through the points $(k-4,\,2k-5)$ and $(3k+2,\,6k+7)$ , where $k$ is a nonzero constant with $k \ne -3$ .

Which of the following equations represents this line?

For line-equation questions with symbolic coordinates, first focus on the slope: plug the coordinates into $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ , simplify carefully, and cancel any common factors (watching for stated restrictions like $k \ne -3$ that justify cancellation). Once you have the slope, immediately write $y = mx + b$ , substitute one of the given points, and solve quickly for $b$ . Finally, match your simplified equation to the options by checking both the slope and the y-intercept, which is faster and less error-prone than expanding or rearranging every answer choice.

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Start with the slope formula

Use the slope formula $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ with the two points $(k-4,\,2k-5)$ and $(3k+2,\,6k+7)$ . Write out the numerator and denominator carefully.

Simplify using algebra with k

Treat $k$ like a regular number. Expand and combine like terms in the numerator and denominator of the slope; look for a common factor that can cancel, especially something involving $(k+3)$ .

Write the line in slope-intercept form

Once you have the slope, write the equation in the form $y = mx + b$ . Then plug in the coordinates of one of the given points to solve for $b$ .

Match with the answer choices

After finding both the slope and y-intercept, compare your equation to the four choices and select the one that has the same slope and y-intercept.

Desmos Guide

Create a specific example using a value of k

In Desmos, define a value for $k$ that satisfies the conditions, for example type k = 1. Then compute the two points: $(k-4, 2k-5)$ becomes $(-3, -3)$ and $(3k+2, 6k+7)$ becomes $(5, 13)$ when $k=1$ .

Plot the example points and candidate lines

Enter the two points as (-3, -3) and (5, 13) using the point tool or by typing them. Then graph each answer choice as separate equations: y = 2x - 3, y = -2x + 3, y = (1/2)x + 3, and y = 2x + 3.

Identify the matching equation

Look at the graph and see which of the four lines passes through both of the plotted points. The equation of that line is the correct choice.

Step-by-step Explanation

Write the slope using the two given points

The slope $m$ of a line through $(x_1,y_1)$ and $(x_2,y_2)$ is

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

Here, $(x_1, y_1) = (k-4,\,2k-5)$ and $(x_2, y_2) = (3k+2,\,6k+7)$ , so

m = \frac{(6k+7) - (2k-5)}{(3k+2) - (k-4)}.

Simplify the slope expression

Simplify the numerator:

$(6k+7) - (2k-5) = 6k+7 -2k +5 = 4k+12 = 4(k+3)$ .

Simplify the denominator:

$(3k+2) - (k-4) = 3k+2 -k +4 = 2k+6 = 2(k+3)$ .

So the slope becomes

m = \frac{4(k+3)}{2(k+3)}.

Since $k \ne -3$ , we can cancel $(k+3)$ to get a numerical value for the slope.

Find the numerical slope and write the general form of the line

Cancel $(k+3)$ in the fraction:

m = \frac{4(k+3)}{2(k+3)} = \frac{4}{2} = 2.

So the line has slope 2. In slope-intercept form $y = mx + b$ , this line is $y = 2x + b$ for some constant $b$ . Next, use one of the given points to solve for $b$ .

Use a point to find the y-intercept and match the equation

Substitute the point $(k-4,\,2k-5)$ into $y = 2x + b$ :

2k - 5 = 2(k - 4) + b.

Compute the right side: $2(k - 4) = 2k - 8$ , so

2k - 5 = 2k - 8 + b.

Subtract $2k$ from both sides: $-5 = -8 + b$ , so $b = 3$ .

Thus, the equation of the line is $y = 2x + 3$ , 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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Desmos Guide

Step-by-step Explanation