A line is shown on the coordinate plane. The line represents the graph of , where is a linear function. Which equation defines ?

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SAT Linear Equations in Two Variables Question 21 | Free SAT Prep | Aniko

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Question 21·Hard·Linear Equations in Two Variables

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A line is shown on the coordinate plane. The line represents the graph of $y = 2f(x) + x$ , where $f$ is a linear function.

Which equation defines $f$ ?

First find the equation of the graphed line by using two points to get the slope and then solving for the intercept in $y=mx+b$ . Next, use the structure $y=2f(x)+x$ to isolate the function: rewrite it as $2f(x)=y-x$ , then substitute your line equation for $y$ and divide by 2.

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Find the slope from the two points

Use the labeled points $A(-7,6)$ and $B(5,-4)$ and compute $m=\frac{y_2-y_1}{x_2-x_1}$ .

Find the y-intercept

After you have the slope, plug one labeled point into $y=mx+b$ to solve for $b$ .

Isolate $f(x)$

Use $y=2f(x)+x$ . Subtract $x$ from both sides to get $2f(x)=y-x$ , then divide by 2.

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Plot the two labeled points

Enter the points from the graph: A=(-7,6) and B=(5,-4).

Find the line equation

Compute the slope with m=(-4-6)/(5-(-7)), then graph the line using point-slope form: y=m(x-5)-4.

Express $f(x)$ from the relationship

Since $y=2f(x)+x$ , enter f(x)=(m(x-5)-4 - x)/2 and simplify the result to match an answer choice.

Step-by-step Explanation

Find the slope of the graphed line

From the graph, the line passes through $A(-7,6)$ and $B(5,-4)$ . The slope is

m=\frac{-4-6}{5-(-7)}=\frac{-10}{12}=-\frac{5}{6}.

Write the graphed line in slope-intercept form

Use $B(5,-4)$ in $y=mx+b$ :

-4=\left(-\frac{5}{6}\right)(5)+b=-\frac{25}{6}+b \implies b=\frac{1}{6}.

So the graphed line is

y=-\frac{5}{6}x+\frac{1}{6}.

Use the relationship $y=2f(x)+x$

Since the line is the graph of $y=2f(x)+x$ , substitute $y=-\frac{5}{6}x+\frac{1}{6}$ :

2f(x)+x=-\frac{5}{6}x+\frac{1}{6}.

Subtract $x$ from both sides:

2f(x)=-\frac{5}{6}x-x+\frac{1}{6}=-\frac{11}{6}x+\frac{1}{6}.

Solve for $f(x)$ and select the choice

Divide by 2:

f(x)=\frac{-\frac{11}{6}x+\frac{1}{6}}{2}=-\frac{11}{12}x+\frac{1}{12}.

Therefore, $f(x) = -\tfrac{11}{12} x + \tfrac{1}{12}$ .

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