Chapter 6.2, Problem 29P

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

$2x=3y-9$

$-3x+y=10$

To determine

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

  $\begin{array}{l}2x=3y-9\\ -3x+y=10\end{array}$

Answer to Problem 29P

The solution is

  $\left(x,y\right)=(-3,1)$

The system has one solution.

Explanation of Solution

Given information:

  $\begin{array}{l}2x=3y-9\\ -3x+y=10\end{array}$

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.

  $\begin{array}{l}2x=3y-9\\ -3x+y=10\end{array}$

First we solve equation $-3x+y=10$ as follows:

  $\begin{array}{l}-3x+y=10\\ \Rightarrow y=10+3x\end{array}$

Now we substitute $y=10+3x$ into $2x=3y-9$ , we get,

  $\begin{array}{l}2x=3y-9\\ \Rightarrow 2x=3\left(10+3x\right)-9\\ \Rightarrow -7x=21\\ \Rightarrow x=-3\end{array}$

By plugging $x=-3$ in equation $y=10+3x$ we get

  $\begin{array}{l}y=10+3x\\ \Rightarrow y=10+3(-3)\\ \Rightarrow y=1\end{array}$

The solution is $\left(x,y\right)=(-3,1)$

The system has one solution.