The table shows three values of and their corresponding values of , where is a constant, for the linear relationship between and . | | | |----|----| | | | | | | | | | What is the slope of the line that represents this relationship in the -plane?

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SAT Linear Equations in Two Variables Question 14 | Free SAT Prep | Aniko

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Question 14·Medium·Linear Equations in Two Variables

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The table shows three values of $x$ and their corresponding values of $y$ , where $n$ is a constant, for the linear relationship between $x$ and $y$ .

$x$	$y$
$-6$	$n + 184$
$-3$	$n + 92$
$0$	$n$

What is the slope of the line that represents this relationship in the $xy$ -plane?

When a table gives a linear relationship involving a constant like $n$ , immediately write the slope formula $m = (y_2 - y_1)/(x_2 - x_1)$ and choose the pair of points that makes the arithmetic simplest. Substitute carefully with parentheses, simplify symbolically, and notice that any $n$ terms in the numerator should cancel, leaving a constant numerical slope; avoid answers that still contain $n$ or that correspond to flipping the fraction (run over rise instead of rise over run).

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Use the definition of slope

Remember that the slope of a line is the change in $y$ divided by the change in $x$ between any two points on the line. Write down the slope formula using two of the table’s points.

Pick convenient points

Choose the two points that look easiest to work with. Which pair gives you simple numbers for the change in $x$ ?

Be careful with subtraction and parentheses

When you compute $y_2 - y_1$ , write the entire first y-value and the entire second y-value in parentheses, especially the one with $n+92$ , so you correctly distribute the minus sign.

Look for cancellation of $n$

After you substitute the $y$ -values, combine like terms in the numerator. What happens to the $n$ terms when you subtract?

Desmos Guide

Assign a specific value to n

In Desmos, type something like n = 0 (or any other simple number). This lets you turn the expressions in the table into actual coordinates, but the slope you find will be the same for any value of $n$ .

Enter two of the points and compute the slope

Using your chosen value of $n$ , write down two corresponding points, for example $(-3, n+92)$ and $(0, n)$ become numeric points. Then in Desmos, type the expression (y2 - y1)/(x2 - x1) using those numbers (for instance, (n - (n+92))/(0 - (-3)) with your chosen $n$ ).

Read the slope from Desmos

Look at the numeric result of the expression you entered; that value is the slope of the line. Choose the answer option that matches this number.

Step-by-step Explanation

Recall the slope formula

For a line that passes through two points $(x_1, y_1)$ and $(x_2, y_2)$ , the slope $m$ is given by:

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

You can choose any two points from the table because the relationship is linear.

Choose the simplest pair of points

Pick the points $(-3,\, n+92)$ and $(0,\, n)$ , because their x-values are close together and the arithmetic is easy.

Let $(x_1, y_1) = (-3,\, n+92)$ and $(x_2, y_2) = (0,\, n)$ .

Now plug these into the slope formula:

m = \frac{n - (n+92)}{0 - (-3)}

Simplify the numerator and denominator symbolically

Simplify the numerator $n - (n+92)$ by distributing the negative sign:

$n - (n+92) = n - n - 92$

The denominator $0 - (-3)$ becomes:

$0 - (-3) = 3$

So the slope expression becomes:

m = \frac{n - n - 92}{3}

Use cancellation to find the numerical slope

Notice that $n - n = 0$ , so the numerator simplifies to just $-92$ :

m = \frac{-92}{3}

So, the slope of the line is $-92/3$ , which corresponds to answer choice A.

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Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation