Chapter 9.8, Problem 3STP

For Exercises 1-4, choose the correct letter.

What is the solution of the system of equations shown below? $y=x^2-5x+18$

$y=4x+4$

1. $(-2,-4)$
2. and $(-7,-24)$

C. $(2,12)$

and $(7,32)$

$(0,4)$

and $(2,12)$

D. $(4,20),$

and $(5,24)$

To determine

To find the solution of the system of equation.

Answer to Problem 3STP

The answer matched with the $\left(\text{c}\right)$

Explanation of Solution

Given information:

The equation is

  $\begin{array}{l}y=x^2-5x+18\\ y=4x+4\end{array}$

The given answer

  $\begin{array}{l}\left(\text{a}\right)\text{. }\left(-2,-4\right)\text{ and }\left(-7,-24\right)\left(\text{b}\right)\text{. }\left(0,4\right)\text{ and }\left(2,12\right)\\ \\ \left(\text{c}\right)\text{. }\left(2,12\right)\text{ and }\left(7,32\right)\left(\text{d}\right)\text{. }\left(4,20\right)\text{ and }\left(5,24\right)\end{array}$

The solution can be obtained as

  $\begin{array}{l}y=x^2-5x+18\left[\text{equation 1}\right]\\ y=4x+4\left[\text{equation 2}\right]\end{array}$

Substitute the value of y from equation 2 in equation 1

  $\begin{array}{l}{x}^2-5x+18=4x+4\\ x^2-5x+18-4x-4=0\\ x^2-9x+14=0\\ x^2-\left(7x+2x\right)+14=0\left[\text{write 9}x=\left(7x+2x\right)\right]\\ x^2-7x-2x+14=0\\ x\left(x-7\right)-2\left(x-7\right)=0\left[\begin{array}{l}\text{the first and second term has a common factor,}x\text{ and}\\ \text{the third and fourth term has a common factor,2}\end{array}\right]\\ \left(x-7\right)\left(x-2\right)=0\end{array}$

Therefore,

  $\begin{array}{l}\left(x-7\right)=0\text{ or }\left(x-2\right)=0\text{ by zero product property}\\ x=7\text{ or }x=2\end{array}$

Solve for the corresponding y value using equation 2

  $\begin{array}{l}y=4x+4\\ \text{put the value of }x=7\\ y=4\left(7\right)+4\\ y=28+4\\ y=32\\ \text{again,put the value of }x=2\\ y=4\left(2\right)+4\\ y=8+4\\ y=12\end{array}$

Hence,

The answer matched with the $\left(\text{c}\right)$