Chapter 12.1, Problem 69E

To determine

To calculate: The value of $a-b$ if $a-4b=15$ and $4a-b=15$ .

Answer to Problem 69E

The value of $a-b$ is $6$ . The correct option is C.

Explanation of Solution

Given information:

The equations $a-4b=15$ and $4a-b=15$ .

The possible options:

  $\begin{array}{l}\text{A}\text{. 3}\\ \text{B}\text{. 4}\\ \text{C}\text{. 6}\\ \text{D}\text{. 15}\\ \text{E}\text{. 20}\end{array}$

Calculation:

Consider the equations $a-4b=15$ and $4a-b=15$ .

Use the method of substitution to solve the system above.

From the equation $a-4b=15$ isolate the value of a,

  $\begin{array}{c}a-4b=15\\ a=15+4b\end{array}$

Now substitute $a=15+4b$ in the equation $4a-b=15$ ,

  $\begin{array}{c}4\left(15+4b\right)-b=15\\ 60+16b-b=15\\ 15b=-45\\ b=-3\end{array}$

Now, substitute $b=-3$ in the equation $a=15+4b$ ,

  $\begin{array}{c}a=15+4\left(-3\right)\\ a=15-12\\ a=3\end{array}$

Therefore, $a-b$ is,

  $\begin{array}{c}3-\left(-3\right)=3+3\\ =6\end{array}$

Thus, the solution set for the system of equations is $\left(-2,-7\right)$ .