To solve the system of equations graphically, we need to plot both equations on the same set of axes.
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First Equation: \( y = \frac{3}{2}x - 3 \)
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This is in slope-intercept form \( y = mx + b \), where \( m = \frac{3}{2} \) is the slope and \( b = -3 \) is the y-intercept.
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To find points for this line, we can calculate the y-value for several x-values:
- If \( x = 0 \): \[ y = \frac{3}{2}(0) - 3 = -3 \quad \text{(point: (0, -3))} \]
- If \( x = 2 \): \[ y = \frac{3}{2}(2) - 3 = 3 - 3 = 0 \quad \text{(point: (2, 0))} \]
- If \( x = 4 \): \[ y = \frac{3}{2}(4) - 3 = 6 - 3 = 3 \quad \text{(point: (4, 3))} \]
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Plot these points (0, -3), (2, 0), and (4, 3) and draw the line.
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Second Equation: \( x + y = 7 \)
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Rearranging this into slope-intercept form gives \( y = -x + 7 \).
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To find points for this line, we can calculate the y-value for several x-values:
- If \( x = 0 \): \[ y = -0 + 7 = 7 \quad \text{(point: (0, 7))} \]
- If \( x = 7 \): \[ y = -7 + 7 = 0 \quad \text{(point: (7, 0))} \]
- If \( x = 3 \): \[ y = -3 + 7 = 4 \quad \text{(point: (3, 4))} \]
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Plot these points (0, 7), (7, 0), and (3, 4) and draw the line.
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Finding the Intersection Point:
- The solution to the system of equations is where the two lines intersect.
- By visually inspecting the plotted lines, we can find the coordinates of the intersection point.
Intersection Calculation:
To find the exact intersection algebraically, we can substitute \( y \) from the first equation into the second equation:
Substitute \( y = \frac{3}{2}x - 3 \) into \( x + y = 7 \):
\[ x + \left(\frac{3}{2}x - 3\right) = 7 \] \[ x + \frac{3}{2}x - 3 = 7 \] \[ \frac{5}{2}x - 3 = 7 \] Add 3 to both sides: \[ \frac{5}{2}x = 10 \] Multiply both sides by \(\frac{2}{5}\): \[ x = 4 \]
Now, substitute \( x = 4 \) back into the first equation to find \( y \): \[ y = \frac{3}{2}(4) - 3 = 6 - 3 = 3 \]
Solution:
The solution to the system of equations is \( (4, 3) \). This point is where the two lines intersect on the graph.
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