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8 A test has 30 questions worth 100 points. The test consists of true/false questions worth 2 points and some multiple choice questions worth 12 points. The number of true/false questions in the test are
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$8$ A test has $30$ questions worth $100$ points. The test consists of true/false questions worth $2$ points and some multiple choice questions worth $12$ points. The number of true/false questions in the test are $\square$

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Counting principle

Full solution

Q.

8

A test has

30

questions worth

100

points. The test consists of true/false questions worth

2

points and some multiple choice questions worth

12

points. The number of true/false questions in the test are

\square

Define Variables: Let $x$ be the number of true/false questions, each worth $2$ points.
Form Equations: Let $y$ be the number of multiple choice questions, each worth $12$ points.
Solve System: The total number of questions is $30$ , so $x + y = 30$ .
Eliminate Variable: The total points for the test are $100$ , so $2x + 12y = 100$ .
Simplify Equation: Now we solve the system of equations: $\newline$ $x + y = 30$ $\newline$ $2x + 12y = 100$
Find Value of y: Multiply the first equation by $2$ to help eliminate $y$ : $2x + 2y = 60$
Substitute to Find x: Subtract the modified first equation from the second equation: $\newline$ $(2x + 12y) - (2x + 2y) = 100 - 60$
Substitute to Find x: Subtract the modified first equation from the second equation: $\newline$ $(2x + 12y) - (2x + 2y) = 100 - 60$ This simplifies to $10y = 40$ .
Substitute to Find x: Subtract the modified first equation from the second equation: $\newline$ $(2x + 12y) - (2x + 2y) = 100 - 60$ This simplifies to $10y = 40$ .Divide both sides by $10$ to find $y$ : $\newline$ $y = \frac{40}{10}$ $\newline$ $y = 4$
Substitute to Find x: Subtract the modified first equation from the second equation: $\newline$ $(2x + 12y) - (2x + 2y) = 100 - 60$ This simplifies to $10y = 40$ .Divide both sides by $10$ to find $y$ : $\newline$ $y = \frac{40}{10}$ $\newline$ $y = 4$ Now substitute $y$ back into the first equation to find $x$ : $\newline$ $x + 4 = 30$ $\newline$ $x = 30 - 4$ $\newline$ $10y = 40$ $0$

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