Chapter 6.2, Problem 31P

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

$x=6y+16$

$9x-2y=-12$

To determine

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

  $\begin{array}{l}x=6y+16\\ 9x-2y=-12\end{array}$

Answer to Problem 31P

The solution is $\left(x,y\right)=(-2,-3)$

The system has one solution.

Explanation of Solution

Given information:

  $\begin{array}{l}x=6y+16\\ 9x-2y=-12\end{array}$

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.

  $\begin{array}{l}x=6y+16\\ 9x-2y=-12\end{array}$

As the equation $x=6y+16$ .

Now we substitute $x=6y+16$ into $9x-2y=-12$ , we get,

  $\begin{array}{l}9x-2y=-12\\ \Rightarrow 9\left(6y+16\right)-2y=-12\\ \Rightarrow 54y+144-2y=-12\\ \Rightarrow 52y=-156\\ \Rightarrow y=-3\end{array}$

By plugging $y=-3$ in equation $x=6y+16$ we get

  $\begin{array}{l}x=6y+16\\ \Rightarrow x=6(-3)+16\\ \Rightarrow x=-2\end{array}$

Therefore the solution is

  $\left(x,y\right)=(-2,-3)$

The system has one solution.