To estimate the solution to the system of linear equations given by \( y = -x + 6 \) and \( y = 3x - 2 \), we should find the point where the two lines intersect.
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Identify the equations:
- The first line, \( y = -x + 6 \), has a y-intercept at (0, 6) and a slope of -1.
- The second line, \( y = 3x - 2 \), has a y-intercept at (0, -2) and a slope of 3.
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Graph the lines:
- For the line \( y = -x + 6 \), if \( x = 0 \), then \( y = 6 \). If \( x = 6 \), then \( y = 0 \) (point (6, 0)). This line goes downwards from left to right.
- For the line \( y = 3x - 2 \), if \( x = 0 \), then \( y = -2 \). If \( x = 2 \), then \( y = 4 \) (point (2, 4)). This line goes upwards from left to right.
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Find the intersection:
- To find where they intersect algebraically, set the equations equal to each other: \[ -x + 6 = 3x - 2 \]
- Rearranging gives: \[ 6 + 2 = 3x + x \] \[ 8 = 4x \] \[ x = 2 \]
- Substitute \( x = 2 \) into one of the original equations to find \( y \): \[ y = -2 + 6 = 4 \]
Thus, the solution to the system of equations, or the point of intersection of the two lines, is \( (2, 4) \).