Solve the system of equations.

y=x2+38x+37
y=50x+1
Write the coordinates in exact form. Simplify all fractions and radicals.
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1 answer

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:

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