The table below shows the coordinates of three points on a line in the -plane, where and are constants. | | | |----:|----:| | 3 | 7 | | | 11 | | 12 | | If the slope of the line is , what is the value of ?

Solution available with step-by-step explanation

SAT Linear Equations in Two Variables Question 18 | Free SAT Prep | Aniko

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Question 18·Hard·Linear Equations in Two Variables

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The table below shows the coordinates of three points on a line in the $xy$ -plane, where $k$ and $n$ are constants.

$x$	$y$
3	7
$k$	11
12	$n$

If the slope of the line is $2$ , what is the value of $k+n$ ?

Your answer

(Express the answer as an integer)

For line-and-slope questions like this, immediately write down the slope formula $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ and set it equal to the given slope using pairs of points that contain the unknowns. Solve the simple linear equations one at a time (first for $k$ , then for $n$ ), and only after both are found, perform the final operation the question asks for (here, adding $k$ and $n$ ). Carefully organize your work so you do not accidentally flip the numerator and denominator in the slope formula, which is a very common source of errors.

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Connect the points and the slope

All three points lie on the same line, and the slope of that line is 2. What must be true about the slope computed between any two of the listed points?

Use the slope formula with k

Write the slope formula between $(3,7)$ and $(k,11)$ : $\dfrac{11 - 7}{k - 3}$ . Set this equal to 2 and solve for $k$ .

Use the slope formula with n

Now write the slope formula between $(3,7)$ and $(12,n)$ : $\dfrac{n - 7}{12 - 3}$ . Set this equal to 2, solve for $n$ , and then add your values of $k$ and $n$ .

Desmos Guide

Graph the line using a point and the slope

Use the point $(3,7)$ and slope 2 to write an equation of the line. In Desmos, type y - 7 = 2(x - 3) and then add another line y = 2x + 1 to see the simplified slope-intercept form; both should overlap as the same line.

Use intersections to find k and n visually

To find $k$ , graph the horizontal line y = 11 and use the intersection tool or click where it meets your line; the $x$ -coordinate of that intersection is $k$ . To find $n$ , graph the vertical line x = 12 and find where it meets your line; the $y$ -coordinate of that intersection is $n$ .

Check the final sum in Desmos

After you have read $k$ and $n$ from the graph, type their sum into Desmos (for example, if you found $k = a$ and $n = b$ , type a + b). The displayed value is what you should report as $k + n$ .

Step-by-step Explanation

Use the slope formula with the first two points to find k

Because all three points are on the same line and the slope is 2, the slope between any two of them is 2.

Use points $(3,7)$ and $(k,11)$ in the slope formula:

\text{slope} = \frac{11 - 7}{k - 3}

Set this equal to 2:

\frac{11 - 7}{k - 3} = 2

Simplify the numerator $11 - 7$ to get 4 and solve for $k$ :

\frac{4}{k - 3} = 2

Multiply both sides by $k - 3$ and then solve the resulting equation for $k$ .

Use the slope formula with the first and third points to find n

Now use points $(3,7)$ and $(12,n)$ with the slope formula. The slope is still 2, so:

\text{slope} = \frac{n - 7}{12 - 3}

Set this equal to 2:

\frac{n - 7}{12 - 3} = 2

Simplify the denominator $12 - 3$ to get 9:

\frac{n - 7}{9} = 2

Multiply both sides by 9 and solve the resulting equation for $n$ .

Add k and n to answer the question

From Step 1, solving $\dfrac{4}{k - 3} = 2$ gives:

\begin{aligned} 4 &= 2(k - 3) \\ 4 &= 2k - 6 \\ 2k &= 10 \\ k &= 5 \end{aligned}

From Step 2, solving $\dfrac{n - 7}{9} = 2$ gives:

\begin{aligned} n - 7 &= 18 \\ n &= 25 \end{aligned}

The question asks for $k + n$ :

k + n = 5 + 25 = 30

So the value of $k + n$ is $30$ .

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Step-by-step Explanation

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