Question

Solve the system of equations.
y=x2+38x+37
y=50x+1
Write the coordinates in exact form. Simplify all fractions and radicals.
,

Answer 1

To solve the system of equations given by:

1. \( y = x^2 + 38x + 37 \) (Equation 1)
2. \( y = 50x + 1 \) (Equation 2)

We can set the two equations equal to each other since they both equal \( y \):

\[
x^2 + 38x + 37 = 50x + 1
\]

Now, we'll rearrange the equation to gather all terms on one side:

\[
x^2 + 38x + 37 - 50x - 1 = 0
\]

Combining like terms gives us:

\[
x^2 - 12x + 36 = 0
\]

Next, we can factor the quadratic equation:

\[
(x - 6)(x - 6) = 0
\]

Or simply:

\[
(x - 6)^2 = 0
\]

This implies that:

\[
x - 6 = 0 \implies x = 6
\]

Now, to find the corresponding \( y \) value, we can substitute \( x = 6 \) back into either of the original equations. It's often easiest to use the linear equation (Equation 2):

\[
y = 50(6) + 1 = 300 + 1 = 301
\]

Thus, the solution to the system of equations, or the coordinates of the intersection point, is:

\[
\boxed{(6, 301)}
\]