Question 95·200 Super-Hard SAT Math Questions·Algebra

0 / 200 Questions completed

The graph of a linear equation passes through the points $\left(-9, \frac{5}{4}\right)$ and $\left(15, -\frac{7}{2}\right)$ . The graph also passes through the point $\left(\frac{3}{2}, a\right)$ . Which choice is the value of $a$ ?

For a linear function, two points determine one line. Turn the points into equations, use a Desmos regression to find the line’s slope and intercept, then evaluate it at the requested input. The first coordinate is the input; the second is the output. A given output is not the requested output unless its input matches.

Hints

Hint 1
A linear function has the form $y=mx+b$ , where $m$ is its constant rate of change and $b$ is its output at $x=0$ . What equations must the two given points satisfy?
Hint 2
A Desmos regression can match both point equations at once. Put their expressions in one list and their known outputs in another, joined by $\sim$ . This finds the slope and intercept of the line.
Hint 3
In $(3/2,a)$ , the first coordinate is the input and the second is the output. Once you know the line’s slope and intercept, what value should you put in for $x$ ?

Step-by-step

Fit the line, then evaluate it
Step 1
Turn the points into equations
Write the line as $y=mx+b$ , where the slope $m$ is the change in $y$ for each increase of $1$ in $x$ , and $b$ is the output at $x=0$ . Each given point must make the equation true, so $\frac54=-9m+b,\qquad -\frac72=15m+b$ .
Step 2
Find the line in Desmos
Type $[-9m+b,15m+b]\sim[5/4,-7/2]$ . The regression symbol $\sim$ tells Desmos to find $m$ and $b$ that match both equations. Under PARAMETERS, it shows $m\approx-0.1979167$ and $b=-0.53125$ .
Step 3
Identify the requested output
In $(3/2,a)$ , the first coordinate is the input and the second is the output. Put $3/2$ into the line, so $a=m\left(\frac32\right)+b$ . The known outputs belong to different inputs, so neither can be used as $a$ .
Step 4
Evaluate the line at the new input
Type $m(3/2)+b$ below the regression. Desmos displays $-0.828125$ ; tap its fraction button to see $-53/64$ . So the output $a$ at $x=3/2$ is $-53/64$ . Choice D.

Question 95·200 Super-Hard SAT Math Questions·Algebra

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this