Chapter 9.7, Problem 23IP

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 23IP

The system of equations representing the given situation:

  $\begin{array}{l}x+y=15\\ \text{ 2}x=3y\end{array}$

The solution of this system of equations is $\text{(9},6)$ .

From the solution of the system of equations, it can be concluded that the two-digits are $9$ and $6$ , so the number is $96$ .

Explanation of Solution

Given information :

The sum of the digits in a two-digit number is $15$ . Three times the second digit equals two times the first digit.

Calculation :

As per problem,

Let the first digitof a two-digit numberbe $x$ and the second digitbe $y$ .

The sum of the digitsis $15$ .Three times the second digit ( $y$ ) equals two times the first digit ( $x$ ).

Thegiven situationcan be represented as the following system of equations:

  $\begin{array}{l}x+y=15\\ \text{ 2}x=3y\end{array}$

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

  $\begin{array}{l}\text{}\text{}\text{}\text{}\text{}\text{}x+y=15\text{ [Write the first equation]}\\ x+y-y=15-y\text{ [Subtract }y\text{ from each side]}\\ x=15-y\text{ [Simplify}]\end{array}$

Find the value of $y$ by substituting $15-y$ for $x$ in the second equation.

  $\begin{array}{l}\text{ 2}x=3y\text{ [Write the second equation]}\\ 2(15-y)=3y\text{ [Substitute }x=15-y\text{]}\\ 30-2y=3y\text{ [Combine like terms}]\\ 30-2y+2y=3y+2y\text{ [Add 2}y\text{ from each side}]\\ 30=5y\text{ [Simplify}]\\ 6=y\text{ [Divide both sides by 5}]\end{array}$

Substitute $6$ for $y$ in either equation to find the value of $x$ .

  $\begin{array}{l}\text{}\text{}\text{}\text{}\text{}x+y=15\text{ [Write the first equation]}\\ x+6=19\text{ [}y=6\text{]}\\ x=9\text{ [Subtract 6 from each side}]\end{array}$

The solution of this system of equations is $\text{(9},6)$ .

The two-digits are $9$ and $6$ , so the number is $96$ .