To solve the system of equations given by \(y = 3x - 2\) and \(y = -x - 6\) graphically, we will find the intersection point of these two lines, which represents the solution to the system.
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Graph the first equation \(y = 3x - 2\):
- This is a line with a slope of 3 and a y-intercept at \(-2\).
- To find points, you can plug in values for \(x\):
- If \(x = 0\), \(y = 3(0) - 2 = -2\) → Point (0, -2)
- If \(x = 1\), \(y = 3(1) - 2 = 1\) → Point (1, 1)
- If \(x = -1\), \(y = 3(-1) - 2 = -5\) → Point (-1, -5)
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Graph the second equation \(y = -x - 6\):
- This is a line with a slope of -1 and a y-intercept at \(-6\).
- To find points, plug in values for \(x\):
- If \(x = 0\), \(y = -0 - 6 = -6\) → Point (0, -6)
- If \(x = 1\), \(y = -1 - 6 = -7\) → Point (1, -7)
- If \(x = -6\), \(y = -(-6) - 6 = 0\) → Point (-6, 0)
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Plot both lines on a graph:
- For \(y = 3x - 2\), plot points (0, -2), (1, 1), (-1, -5) and draw the line through them.
- For \(y = -x - 6\), plot points (0, -6), (1, -7), (-6, 0) and draw the line through them.
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Find the intersection point of the two lines:
- From the equations, you can set them equal to find \(x\): \[ 3x - 2 = -x - 6 \] Combine like terms: \[ 3x + x = -6 + 2 \Rightarrow 4x = -4 \Rightarrow x = -1 \]
- Substitute \(x = -1\) back into either equation to find \(y\): \[ y = 3(-1) - 2 = -3 - 2 = -5 \] or, \[ y = -(-1) - 6 = 1 - 6 = -5 \]
Thus, the intersection point is \((-1, -5)\).
Conclusion: The estimated solution to the system of equations based on the graph is \((-1, -5)\).