Q46. - Algebra 2

Question

Mr. Talbot is writing a test for his science classes. The test will have true/false questions worth 2 points each and multiple-choice questions worth 4 points each for a total of $100$ points. He wants to have twice as many multiple-choice questions as true/false.

How many true/false questions and multiple-choice questions will be on the test?

Step-by-Step Solution

Answer

There are $10$ true/false questions and $20$ multiple-choice questions.

Step-1 – Apply the concept of system of equations

A set of two or more equations with the same variables is called a system of linear equations. The ordered pairs that satisfy all the equations in the system is called as solution of the equations.

Step-2 – Formation of the equations

Here, x represents the number of true/false questions and y represents the number of multiple-choice questions. The equations so formed according to the question are:

$\begin{array}{c}2x+4y=100\\ y=2x\end{array}$

Step-3 – Apply the substitution method of solving equations

The algebraic method of substitution involves solving the one of the two equations for one variable in terms of other variable and then substituting the expression so formed for the variable in the second equation.

Step-4 – Substitute the expression

Now, substitute $y=2x$ in the equation $2x+4y=100$ and solve.

$\begin{array}{c}2x+4y=100\\ 2x+4\left(2x\right)=100\\ 2x+8x=100\\ 10x=100\end{array}$

Simplify it further as.

$\begin{array}{c}10x=100\\ x=10\end{array}$

Thus, the value of x is $10$.

Step-5 – Substitute the value of variable

To find the value of y, substitute $x=10$ in the equation $y=2x$ and then solve as shown.

$\begin{array}{c}y=2x\\ y=2\left(10\right)\\ y=20\end{array}$

Thus, the value of y is $20$.

Hence, there are $10$ true/false questions and $20$ multiple-choice questions.