Question

In the following exercises, solve the system of equations by using substitution

$\left\{\begin{array}{c}9x^2+y^2=9\\ y=3x+3\end{array}\right.$

Short answer

The solutions of the system are$(0,3),(-1,0)$.

Step-by-step solution

Step 1. Given information

The given system of equations is$\left\{\begin{array}{c}9x^2+y^2=9\\ y=3x+3\end{array}\right.$

Step 2. Substitution

• Substitute $y=3x+3$into $9x^2+y^2=9$ and solve for x.

$\begin{array}{rcl}9x^2+{(3x+3)}^2& =& 99x^2+9x^2+9+18x=9\\ 18x^2+18& =& 0\\ 18x(x+1)& =& 0\end{array}$

• So, the values ofxare$0,-1$.

Step 3. Solve for y

• Substitute $x=0$into $y=3x+3$ and solve for y.

$\begin{array}{rcl}y& =& 3\times 0+3\\ & =& 3\end{array}$

• Substitute $x=-1$into $y=3x+3$ and solve for y.

$\begin{array}{rcl}y& =& 3\times -1+3\\ & =& 0\end{array}$

• So, the solutions of the system are$(0,3),(-1,0)$