Chapter 6.2, Problem 27P

Solve each system by substitution. Tell whether the system has one solution, infinitely many solutions, or no solution.

$3x+6y=18$

$3y=-\frac{3}{2}x+9$

To determine

To solve: the system by substitution and suggest whether the system has one solution, infinitely many solutions, or no solution.

  $\begin{array}{l}3x+6y=18\\ 3y=-\frac{3}{2}x+9\end{array}$

Answer to Problem 27P

There exist infinitely many solutions.

Explanation of Solution

Given information:

  $\begin{array}{l}3x+6y=18\\ 3y=-\frac{3}{2}x+9\end{array}$

Calculation:

In order to solve two system by substitution, we solve one of the equations for one of the

Variables, and plug this into other equation.

  $\begin{array}{l}3x+6y=18\\ 3y=-\frac{3}{2}x+9\end{array}$

First we solve equation $3y=-\frac{3}{2}x+9$ as follows:

  $\begin{array}{l}3y=-\frac{3}{2}x+9\\ \Rightarrow y=-\frac{1}{2}x+3\end{array}$

Solve the second equation we get

  $\begin{array}{l}6y=18-3x\\ y=3-\frac{1}{2}x\\ y=-\frac{1}{2}x+3\end{array}$

As both the equations are same there exist infinitely many solutions.