Chapter 9, Problem 43CR

To determine

To write and solve the system of equations.

Answer to Problem 43CR

The system of equations is written as:

  $\begin{array}{l}\text{x}-3\text{y}=11\\ \text{x}+\text{y}=1\end{array}$

The solution for the system of equations is $\left(\frac{7}{2},-\frac{5}{2}\right)$ .

The two numbers are $\frac{7}{2}\text{ and}-\frac{5}{2}$ .

Explanation of Solution

Given information:

The given statement is:

“One number subtracted from three times another number is $11$ . The sum of the numbers is $1$ . Write and solve a system of equations to represent this situation. Interpret the solution.”

Calculation :

Let the numbers be $\text{x and y}$ .

The system of equations is written as:

  $\begin{array}{l}\text{x}-3\text{y}=11\\ \text{x}+\text{y}=1\end{array}$

Find the value of $\text{y}$ in terms of $\text{x}$ from the second equation:

  $\begin{array}{l}\text{x}+\text{y}=1\\ \text{y}=-\text{x}+1\left[\text{Subtracting both sides by x}\right]\end{array}$

Substitute the value of $\text{y}=-\text{x}+1$ in the first equation:

  $\begin{array}{l}\text{x}-3\text{y}=11\\ \text{x}-3\text{y}=11\\ \text{x}-3\left(-\text{x}+1\right)=11\\ \text{x}+3\text{x}-3=11\\ 4\text{x}-3=11\\ 4\text{x}=14\\ \text{x}=\frac{14}{4}\\ \text{x}=\frac{7}{2}\end{array}$

Calculate the value of $\text{y}$ substituting the value of $\text{x}$ :

  $\begin{array}{l}\text{y}=-\text{x}+1\\ \text{y}=-\frac{7}{2}+1\\ \text{y}=\frac{-7+2}{2}\\ \text{y}=\frac{-5}{2}\end{array}$

Hence, the solution for the system of equations is $\left(\frac{7}{2},-\frac{5}{2}\right)$ .

The two numbers are $\frac{7}{2}\text{ and}-\frac{5}{2}$ .