Chapter 9.8, Problem 3LC

To determine

To solve given equations using substitution.

Answer to Problem 3LC

Thesolutions are (4, 1) (1, 4)

Explanation of Solution

Given information :

Given equations,

  $\begin{array}{l}y+x=5\\ y=x^2-6x+9\end{array}$

In the given equations, we shall substitute the value of one variable into other equation.

Given equations are,

  $\begin{array}{l}y+x=5\\ y=x^2-6x+9\end{array}$

Subtituting the value of $y$ we get,

  $\begin{array}{l}\left(x^2-6x+9\right)+x=5\\ x^2-6x+9+x-5=0\\ x^2-5x+4=0\end{array}$

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,

  $\begin{array}{l}x=\frac{5\pm \sqrt{{\left(-5\right)}^2-4\left(4\right)}}{2\left(1\right)}\\ x=\frac{5\pm \sqrt{9}}{2}\\ x=4,1\end{array}$

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

We get $y=1,4$

Hence the solutions are (4, 1) (1, 4)