To estimate the solution to the system of equations given by \( y = -x + 6 \) and \( y = 3x - 2 \), we can find the intersection point of the two lines represented by these equations. This intersection point is the solution to the system of equations.
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Set the equations equal to each other since they both equal \( y \): \[ -x + 6 = 3x - 2 \]
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Solve for \( x \):
- Add \( x \) to both sides: \[ 6 = 4x - 2 \]
- Add \( 2 \) to both sides: \[ 8 = 4x \]
- Divide by \( 4 \): \[ x = 2 \]
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Substitute \( x = 2 \) back into one of the original equations to find \( y \). We'll use \( y = -x + 6 \): \[ y = -2 + 6 = 4 \]
Thus, the solution to the system of equations is \( (2, 4) \).
You can verify this by substituting \( x = 2 \) into the second equation \( y = 3x - 2 \): \[ y = 3(2) - 2 = 6 - 2 = 4 \]
So, the intersection point, and therefore the solution to the system of equations, is \( (2, 4) \).