Chapter 3.5, Problem 52E

To determine

To Write: The non-linear system that has two different solution but the $y$ coordinate is same. Sketch the graph and solve.

Answer to Problem 52E

  $(1,3),(5,3)$

Explanation of Solution

Given: The non-linear system that has two different solution but the $y$ coordinate is same.

Consider the points be $(1,3),(5,3)$ .

Then

The system of equation be

  $f(x)=\{\begin{array}{l}{(}^x+(y-3)=4\\ \frac{1}{4}{(}^x+2\end{array}$

Solving the system of equations we get

  $\begin{array}{}$ {(x-3)}^2+\left(\frac{1}{4}{(x-3)}^2+2-3\right)=4$$ {(x-3)}^2+\frac{1}{16}{(x-3)}^4-\frac{1}{2}{(x-3)}^2+1=4$$ {(x-3)}^2+4=\pm 8$$ {(x-3)}^2=-4\pm 8$$ \text{Neglecting(-8)}$$ {(x-3)}^2=-4+8$$ \begin{array}{l}x-3=\pm 2\\ now\\ x_1=3-2=1\\ x_2=3+2=5\\ thus\end{array}$$ y=\frac{1}{4}{(x-3)}^2+2$$ \begin{array}{l}y=\frac{1}{4}(4)+2=3\\ therefore\\ (1,3),(5,3)\end{array}$\end{array}$

Hence, the solutions are $(1,3),(5,3)$ .

The graph of the functions is shown below.