Chapter 9.7, Problem 13IP

To determine

To solve : the given system of equations.

Answer to Problem 13IP

The given system of equations has infinite number of solutions.

Explanation of Solution

Given information :

The given system of equations:

  $\begin{array}{l}2x=\frac{7}{2}+\frac{-3}{2}y\\ 4x+3y=7\end{array}$

Calculation :

As per problem,

Consider the system of equations:

  $\begin{array}{l}2x=\frac{7}{2}+\frac{-3}{2}y\\ 4x+3y=7\end{array}$

Solve the first equation for $x$ .

  $\begin{array}{l}2x=\frac{7}{2}+-\frac{3}{2}y\text{ [Write the first equation]}\\ 2x=\frac{7}{2}-\frac{3}{2}y\text{ [Simplify the equation]}\\ 2x=\frac{7-3y}{2}\text{ [Take LCM of right side]}\\ x=\frac{7-3y}{4}\text{ [Divide each side by 2}]\end{array}$

Substitute $\frac{7-3y}{4}$ for $x$ in the second equation.

  $\begin{array}{l}\text{ 4}x+3y=7\text{ [Write the second equation]}\\ 4\left(\frac{7-3y}{4}\right)+3y=7\text{ [Substitute }x=\frac{7-3y}{4}\text{]}\\ 7=7\text{ [Simplify}]\end{array}$

  $7=7$ is a true statement. The system of equations has infinite number of solutions.