Chapter 9.7, Problem 21IP

To determine

To represent : the given situation by a system of equations and solve it.

To interpret the solution of the system of equations.

Answer to Problem 21IP

The system of equations representing the given situation:

  $\begin{array}{l}x+2y=30\\ x+y=19\end{array}$

The solution of the system of equations is $\text{(8},11)$ .

From the solution of the system of equations, it can be concluded that one number is $8$ and the other number is $11$ .

Explanation of Solution

Given information :

One number added to twice another number is $30$ . The sum of the numbers is $19$ .

Calculation :

As per problem,

Let one of the numbers be $x$ and the other number be $y$ .

So, $x$ added to twice $y$ is $30$ . The sum of the numbers is $19$ .

Consider the system of equations for the given situation:

  $\begin{array}{l}x+2y=30\\ x+y=19\end{array}$

Solve the second equation for $x$ since its coefficient is $1$ .

  $\begin{array}{l}\text{}\text{}\text{}\text{}\text{}\text{}x+y=19\text{ [Write the second equation]}\\ x+y-y=19-y\text{ [Subtract }y\text{ from each side]}\\ x=19-y\text{ [Simplify}]\end{array}$

Find the value of $y$ by substituting $19-y$ for $x$ in the first equation.

  $\begin{array}{l}x+2y=30\text{ [Write the first equation]}\\ \text{19}-y+2y=30\text{ [Substitute }x=19-y\text{]}\\ 19+y=30\text{ [Combine like terms}]\\ 19+y-19=30-19\text{ [Subtract 19 from each side}]\\ y=11\text{ [Simplify}]\end{array}$

Substitute $11$ for $y$ in either equation to find the value of $x$ .

  $\begin{array}{l}\text{}\text{}\text{}\text{}\text{}x+y=19\text{ [Write the second equation]}\\ x+11=19\text{ [}y=11\text{]}\\ x=8\text{ [Subtract 11 from each side}]\end{array}$

The solution of this system of equations is $\text{(8},11)$ .

Thus, one number is $8$ and the other number is $11$ .