To solve the system of equations, we have the two equations:
- \( y = -15x + 47 \)
- \( y = x^2 - 15x + 11 \)
We can set the two expressions for \( y \) equal to each other:
\[ -15x + 47 = x^2 - 15x + 11 \]
Now, simplify this equation. First, we notice that the \( -15x \) terms on both sides can be eliminated:
\[ 47 = x^2 + 11 \]
Next, we rearrange the equation:
\[ x^2 + 11 - 47 = 0 \] \[ x^2 - 36 = 0 \]
Now, factor the equation:
\[ (x - 6)(x + 6) = 0 \]
This gives us:
\[ x - 6 = 0 \quad \text{or} \quad x + 6 = 0 \] \[ x = 6 \quad \text{or} \quad x = -6 \]
Next, we substitute these \( x \) values back into one of the original equations to find the corresponding \( y \) values. We'll use the first equation, \( y = -15x + 47 \).
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For \( x = 6 \): \[ y = -15(6) + 47 = -90 + 47 = -43 \] So one solution is \( (6, -43) \).
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For \( x = -6 \): \[ y = -15(-6) + 47 = 90 + 47 = 137 \] So another solution is \( (-6, 137) \).
Thus, the solutions to the system of equations are:
\[ (6, -43) \quad \text{and} \quad (-6, 137) \]
So the coordinates in exact form are \( \boxed{(6, -43)} \) and \( \boxed{(-6, 137)} \).