Question

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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Answer 1

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 \( x = 1 \):

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

2. 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:

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