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A test has 22 problems, which are worth a total of 113 points. There are two types of problems in the test. Each multiple-choice problem is worth 4 points, and each short-answer problem is worth 9 points.

Write and solve a system equation to answer the following questions.

This test has

multiple-choice problems and

short-answer problems.

Solution

Assume the test has $x$ multiple-choice problems, and $y$ short-answer problems.

It’s given that there are a total of 22 problems, so we have:

$x+y=22$

Since each multiple-choice problem is worth 4 points, $x$ multiple-choice problems would be worth $4x$ points.

Similarly, since each short-answer problem is worth 9 points, $y$ short-answer problems would be worth $9y$ points.

It’s given that all problems in the test are worth a total of 113 points, so we have:

$4x+9y=113$

Now we have a system of equations:

$\begin{cases} x+y = 22 \\ 4x+9y=113 \end{cases}$

We will use substitution to solve this problem. First, we solve for $y$ in the first equation:

$\begin{aligned} x+y\amp=22\\ x+y\subtractright{x}\amp=22\subtractright{x}\\ y\amp=22-x \end{aligned}$

Next, we substitute $y$ with $22-x$ in $4x +9y = 113\text{,}$ and we have:

$\begin{aligned} 4x +9y \amp= 113\\ 4x +9(\substitute{-x+22}) \amp= 113\\ 4x -9x+198 \amp= 113\\ 4x-9x +198 \amp= 113\\ -5x +198 \subtractright{198} \amp= 113 \subtractright{198}\\ -5x \amp= -85\\ x \amp= {17} \end{aligned}$

Finally, we substitute $x={17}$ into $y = 22-x\text{,}$ and we have:

$\begin{aligned} y \amp= 22-x\\ y \amp= 22-\substitute{{17}}\\ y \amp= {5} \end{aligned}$

The solution to the system of equations is $x={17}\text{,}$ $y={5}\text{.}$ This implies there are 17 multiple-choice questions and 5 short-answer questions in the test.