To solve the system of equations given by:
- \( y = -14x + 13 \)
- \( y = x^2 - 14x - 23 \)
we can set the two equations for \( y \) equal to each other:
\[ -14x + 13 = x^2 - 14x - 23 \]
Next, we can simplify this equation. First, we can eliminate \(-14x\) from both sides:
\[ 13 = x^2 - 23 \]
Now, add \( 23 \) to both sides:
\[ 13 + 23 = x^2 \] \[ 36 = x^2 \]
Next, we take the square root of both sides to solve for \( x \):
\[ x = 6 \quad \text{or} \quad x = -6 \]
Now that we have the \( x \) values, we can substitute them back into either equation to find the corresponding \( y \) values. We'll use the first equation \( y = -14x + 13 \):
- For \( x = 6 \):
\[ y = -14(6) + 13 = -84 + 13 = -71 \] So we have the point \((6, -71)\).
- For \( x = -6 \):
\[ y = -14(-6) + 13 = 84 + 13 = 97 \] So we have the point \((-6, 97)\).
Thus, the solutions to the system of equations are:
\[ (6, -71) \quad \text{and} \quad (-6, 97) \]
In exact form, the coordinates are:
\[ \boxed{(6, -71)} \quad \text{and} \quad \boxed{(-6, 97)} \]
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