Chapter 9.8, Problem 16PPE

To determine

To solve given equations using elimination.

Answer to Problem 16PPE

  $\begin{array}{l}x=4,\frac{1}{2}\\ y=-11,-7.5\end{array}$

Explanation of Solution

Given information :

Given equations,

  $\begin{array}{l}y=-x-7\\ y=x^2-4x-5\end{array}$

In the given equations, we shall subtract one equation from other in order to eliminate one variable.

Given equations are,

  $\begin{array}{l}y=-x-7\\ y=x^2-4x-5\end{array}$

Subtract both equations we get,

  $\begin{array}{l}y-y=\left(-x-7\right)-\left(x^2-4x-5\right)\\ 0=-x-7-x^2+4x+5\\ x^2-3x+2=0\end{array}$

Now we know that the solution of a standard quadratic equation $a{x}^2+bx+c=0$ is given by $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Hence the solution of the given equation is,

  $\begin{array}{l}x=\frac{3\pm \sqrt{{\left(3\right)}^2-4\left(2\right)}}{2\left(2\right)}\\ x=\frac{3\pm \sqrt{1}}{4}\\ x=4,\frac{1}{2}\end{array}$

Now put both values of $x$ in any give equation to determine the values of $y$ ,

We get $y=-11,-7.5$