To graph the equations \( y = 4x + 3 \) and \( y = -x - 2 \), we can first find their points of intersection.
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Finding the Intersection: Set the two equations equal to each other: \[ 4x + 3 = -x - 2 \]
Now, solve for \( x \): \[ 4x + x = -2 - 3 \ 5x = -5 \ x = -1 \]
Now substitute \( x = -1 \) back into either equation to find \( y \). Using \( y = 4x + 3 \): \[ y = 4(-1) + 3 \ y = -4 + 3 \ y = -1 \]
So, the coordinates of the intersection point are: \[ (-1, -1) \]
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Finding Additional Points: You can also find points on each line to help with graphing.
For \( y = 4x + 3 \):
- Let \( x = 0 \): \[ y = 4(0) + 3 = 3 \quad \Rightarrow \quad (0, 3) \]
- Let \( x = 1 \): \[ y = 4(1) + 3 = 7 \quad \Rightarrow \quad (1, 7) \]
For \( y = -x - 2 \):
- Let \( x = 0 \): \[ y = -0 - 2 = -2 \quad \Rightarrow \quad (0, -2) \]
- Let \( x = 1 \): \[ y = -1 - 2 = -3 \quad \Rightarrow \quad (1, -3) \]
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Prepare to Graph: Now you have points from both lines:
- Line 1 (\( y = 4x + 3 \)): \( (0, 3), (1, 7), (-1, -1) \)
- Line 2 (\( y = -x - 2 \)): \( (0, -2), (1, -3), (-1, -1) \)
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Graphing: Plot the points on a coordinate plane:
- \( (0, 3) \) and \( (1, 7) \) for \( y = 4x + 3 \), and draw the line through these points.
- \( (0, -2) \) and \( (1, -3) \) for \( y = -x - 2 \), and draw the line through these points.
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Conclusion: The coordinates of the intersection point are \( (-1, -1) \). You can also graph both equations and observe their slopes and intercepts to fully understand their relationship.