Chapter 9.7, Problem 7GP

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 7GP

The system of equations representing the given situation:

  $\begin{array}{l}x+y=6\\ 2x+3y=13\end{array}$

The solution of the system of equations is $\text{(5},1)$ .

From the solution of the system of equations, it can be concluded that Lola bought $5$ $\$2$ cards (represented by $x$ in the equations) and $1$ $\$3$ card (represented by $y$ in the equations) at $\$13$ .

Explanation of Solution

Given information :

Lola paid $\$13$ for $6$ greeting cards. Some cards cost $\$2$ each and some cost $\$3$ each.

Calculation :

As per problem,

Let the number of $\$2$ cards bought by Lola be $x$ and that of $\$3$ cards be $y$ .

Total card bought by Lola is $6$ , and the amount paid for $6$ greeting cards is $\$13$ .

Consider the system of equations for the given situation:

  $\begin{array}{l}x+y=6\\ 2x+3y=13\end{array}$

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

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

Find the value of $y$ by substituting $6-y$ for $x$ in the second equation.

  $\begin{array}{l}\text{ 2}x+3y=13\text{ [Write the second equation]}\\ \text{2(}6-y)+3y=13\text{ [Substitute }x=6-y\text{]}\\ 12-2y+3y=13\text{ [Distributive property}]\\ 12+y=13\text{ [Combine like terms}]\\ 12+y-12=13-12\text{ [Subtract 12 from each side}]\\ y=1\text{ [Simplify}]\end{array}$

Substitute $1$ for $y$ in either equation to find the value of $x$ .

  $\begin{array}{l}\text{}\text{}\text{}\text{}\text{}x+y=6\text{ [Write the first equation]}\\ x+1=6\text{ [}y=1\text{]}\\ x=6\text{ [Subtract 1 from each side}]\end{array}$

The solution of this system of equations is $\text{(5},1)$ .

Thus, Lola bought $5$ $\$2$ cards (represented by $x$ in the equations) and $1$ $\$3$ card (represented by $y$ in the equations)at $\$13$ .