Chapter 9, Problem 24PT

To determine

To solve the system of equations.

Answer to Problem 24PT

The value of pairs of $\left(x,y\right)$ is $\left(-2,-3\right)$ .

Explanation of Solution

Given information :

The equations provided are $y=1-x^2$ and $y=4x+5$ .

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 :

  $y=1-x^2$

Let this be equation 1.

  $y=4x+5$

Let this be equation 2.

  $4x+5=1-x^2$

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

  $=>4x+5-1+x^2=0$

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

  $=>x^2+4x+4=0$

This resulting equation can be simplified by factoring.

  $=>x^2+2x+2x+4=0$

Breaking the middle term.

  $=>x\left(x+2\right)+2\left(x+2\right)=0$

Arranging the common factors.

  $=>\left(x+2\right)\left(x+2\right)=0$

  $=>{\left(x+2\right)}^2=0$

The two factors of the function.

Now, the root is:

  $x+2=0$

Using zero-product property.

  $x=-2$

Put this value of x in equation 2 to get the value of y.

  $y=4\left(-2\right)+5$

Putting $x=-2$ in equation 2.

  $=-8+5$

Simplifying

  $=-3$

Thus the pair of solution is $\left(-2,-3\right)$ .