To graph the equations \(y = x + 2\) and \(y = 3x - 2\), follow these steps:
Step 1: Graph \(y = x + 2\)
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Identify the Slope and Y-Intercept:
- The equation is in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
- Here, the slope \(m = 1\) and the y-intercept \(b = 2\).
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Plot the Y-Intercept:
- Start by plotting the point \((0, 2)\) on the graph.
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Use the Slope to Find Another Point:
- From \((0, 2)\), use the slope to find another point. Since the slope is 1, move 1 unit up (rise) and 1 unit right (run) to get to \((1, 3)\).
- Plot this point.
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Draw the Line:
- Draw a straight line through the points \((0, 2)\) and \((1, 3)\). Extend the line in both directions.
Step 2: Graph \(y = 3x - 2\)
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Identify the Slope and Y-Intercept:
- For this equation, the slope \(m = 3\) and the y-intercept \(b = -2\).
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Plot the Y-Intercept:
- Plot the point \((0, -2)\) on the graph.
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Use the Slope to Find Another Point:
- From \((0, -2)\), use the slope to find another point. Since the slope is 3, move 3 units up (rise) and 1 unit right (run) to get to \((1, 1)\).
- Plot this point.
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Draw the Line:
- Draw a straight line through the points \((0, -2)\) and \((1, 1)\). Extend the line in both directions.
Step 3: Find the Intersection Point
To find the intersection point of the two lines, set the equations equal to each other:
\[ x + 2 = 3x - 2 \]
- Rearranging gives:
\[ 2 + 2 = 3x - x \] \[ 4 = 2x \] \[ x = 2 \]
- Substitute \(x = 2\) back into either equation to find \(y\):
\[ y = 2 + 2 = 4 \]
So, the intersection point is \((2, 4)\).
Summary of the Graphing Process:
- First Line: Start at \((0, 2)\), slope is 1, passes through \((1, 3)\).
- Second Line: Start at \((0, -2)\), slope is 3, passes through \((1, 1)\).
- Intersection Point: The lines intersect at \((2, 4)\).
Finally, make sure to label the lines and the intersection point on your graph for clarity!
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