Question

Solve the system of equations.
y=
–
12x–43
y=x2–30x+38
Write the coordinates in exact form. Simplify all fractions and radicals.
69+
,

Answer 1

To solve the system of equations given by:

1. \( y = -12x - 43 \)
2. \( y = x^2 - 30x + 38 \)

we can set these two equations equal to each other since both are equal to \( y \):

\[
-12x - 43 = x^2 - 30x + 38
\]

Rearranging this equation gives us:

\[
0 = x^2 - 30x + 38 + 12x + 43
\]

Combine the like terms:

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

Now, we can rearrange this into standard quadratic form:

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

Next, we can factor this quadratic:

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

Setting the factor equal to zero gives us:

\[
x - 9 = 0 \implies x = 9
\]

Now, we substitute \( x = 9 \) back into the first equation to find \( y \):

\[
y = -12(9) - 43
\]

Calculating the right-hand side:

\[
y = -108 - 43 = -151
\]

Thus, we have found one solution to the system of equations:

\[
(x, y) = (9, -151)
\]

So the coordinates in exact form are:

\[
\boxed{(9, -151)}
\]