Solve the system of equations.

y=3x+25
y=13x2+3x+12
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. \( y = 3x + 25 \)
  2. \( y = 13x^2 + 3x + 12 \)

we can set the two expressions for \( y \) equal to each other:

\[ 3x + 25 = 13x^2 + 3x + 12 \]

Next, we will simplify this equation by subtracting \( 3x + 25 \) from both sides:

\[ 0 = 13x^2 + 3x + 12 - 3x - 25 \]

This simplifies to:

\[ 0 = 13x^2 - 13 \]

Now, we can factor out the common term:

\[ 0 = 13(x^2 - 1) \]

Next, we set each factor equal to zero:

\[ 13 = 0 \quad \text{(not applicable)} \] \[ x^2 - 1 = 0 \]

We solve \( x^2 - 1 = 0 \):

\[ x^2 = 1 \]

Taking the square root of both sides, we find:

\[ x = 1 \quad \text{or} \quad x = -1 \]

Now we will find the corresponding \( y \)-coordinates for each \( x \) value using the first equation:

  1. For \( x = 1 \):

\[ y = 3(1) + 25 = 3 + 25 = 28 \]

  1. For \( x = -1 \):

\[ y = 3(-1) + 25 = -3 + 25 = 22 \]

Thus, the coordinates of the solutions are:

\[ (1, 28) \quad \text{and} \quad (-1, 22) \]

Therefore, the solutions to the system of equations are:

\[ \boxed{(1, 28)} \quad \text{and} \quad \boxed{(-1, 22)} \]