Chapter 9, Problem 59SGR

To determine

To solve the system of equations.

Answer to Problem 59SGR

The value of pairs of $\left(x,y\right)$ are $\left(\frac{3}{4},\frac{9}{4}\right)$ and $\left(-2,5\right)$ .

Explanation of Solution

Given information :

The equations provided are $y=x^2-4x+9$ and $y=2x$ .

Formula used :

Since there are two equations provided for y, equate the components of both with each other. This will result in a quadratic equation. Solve this equation for x in order to find the solution

The resulting equation can be solved by factoring. To do so, break up the midpoint to arrive at two factors. Then use the zero-product property to equate each factor in order to get the roots.

Calculation :

  $y=x^2-4x+9$

Let this be equation 1.

  $y=2x$

Let this be equation 2.

  $x^2-4x+9=2x$

Since both are equal to y , equate them to get a resulting equation for x

  $=>x^2-4x+9-2x=0$

Bringing the right-hand side to left-hand side.

  $=>x^2-6x+9=0$

This resulting equation can be simplified by factoring.

  $=>x^2-3x-3x+9=0$

Breaking the middle term.

  $=>x\left(x-3\right)-3\left(x-3\right)=0$

Arranging the common factors.

  $=>\left(x-3\right)\left(x-3\right)=0$

The two factors of the function.

Now, the first root is:

  $x-3=0$

Using zero-product property.

  $x=3$

Now, put this value of x in equation 2 to get the value of y.

  $y=2\left(3\right)$

Putting $x=3$ in equation 2.

  $=6$

Thus, the pair of solution is $\left(3,6\right)$ .