Question

Use the substitution method to determine whether the linear-quadratic system y=x^2−9x+39 , y=9x−42 has 0, 1, or 2 solutions.(1 point) This system has solution(s)

Answer 1

To determine the number of solutions for the system of equations using the substitution method, we will substitute \( y \) from the linear equation into the quadratic equation.

The given equations are:
1. \( y = x^2 - 9x + 39 \)
2. \( y = 9x - 42 \)

First, we substitute \( y \) from the second equation into the first equation:

\[
9x - 42 = x^2 - 9x + 39
\]

Next, we rearrange this equation to set it to zero:

\[
0 = x^2 - 9x + 39 - 9x + 42
\]

Combine like terms:

\[
0 = x^2 - 18x + 81
\]

Now we can rewrite this as:

\[
x^2 - 18x + 81 = 0
\]

Next, we will factor the quadratic equation:

\[
(x - 9)(x - 9) = 0
\]

This can be rewritten as:

\[
(x - 9)^2 = 0
\]

This equation has a double root at \( x = 9 \).

Now we can find the corresponding value of \( y \) by substituting \( x = 9 \) back into one of the original equations. We will use the linear equation:

\[
y = 9(9) - 42 = 81 - 42 = 39
\]

So, the point of intersection (or the solution to the system) is \( (9, 39) \).

This system has exactly **1 solution** which occurs at the point \( (9, 39) \). Thus, the answer is that this system has **1 solution**.