To estimate the solution to the system of equations given by the two lines \( y = -x + 6 \) and \( y = 3x - 2 \), we can find the intersection point by setting the equations equal to each other:
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Set the equations equal to each other: \[ -x + 6 = 3x - 2 \]
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Solve for \( x \): \[ 6 + 2 = 3x + x \] \[ 8 = 4x \] \[ x = 2 \]
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Substitute \( x = 2 \) back into either equation to find \( y \). We'll use the first equation: \[ y = -2 + 6 = 4 \]
Therefore, the solution to the system of equations, which is the point of intersection of the two lines, is: \[ (x, y) = (2, 4) \]
This means the lines intersect at the point \( (2, 4) \).