To graph the system of equations given, we will analyze each equation. The equations are:
- \( y = \frac{2}{3}x - 2 \) (blue line)
- \( y = -\frac{1}{2}x + 5 \) (red line)
Step 1: Determine Points for Each Equation
For the blue line \( y = \frac{2}{3}x - 2 \):
To find points, we can select a few values for \( x \) and calculate \( y \):
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When \( x = 0 \): \[ y = \frac{2}{3}(0) - 2 = -2 \quad \Rightarrow \text{point } (0, -2) \]
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When \( x = 3 \): \[ y = \frac{2}{3}(3) - 2 = 0 \quad \Rightarrow \text{point } (3, 0) \]
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When \( x = 6 \): \[ y = \frac{2}{3}(6) - 2 = 2 \quad \Rightarrow \text{point } (6, 2) \]
For the red line \( y = -\frac{1}{2}x + 5 \):
Again, select values for \( x \):
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When \( x = 0 \): \[ y = -\frac{1}{2}(0) + 5 = 5 \quad \Rightarrow \text{point } (0, 5) \]
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When \( x = 6 \): \[ y = -\frac{1}{2}(6) + 5 = 2 \quad \Rightarrow \text{point } (6, 2) \]
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When \( x = 10 \): \[ y = -\frac{1}{2}(10) + 5 = 0 \quad \Rightarrow \text{point } (10, 0) \quad (\text{but this is outside the graph range: max y = 8}) \]
Summary of Points:
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Blue line points:
- (0, -2)
- (3, 0)
- (6, 2)
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Red line points:
- (0, 5)
- (6, 2) (This point is also on the blue line.)
Step 2: Finding the Intersection Point
To find the intersection point, we can set the two equations equal to each other:
\[ \frac{2}{3}x - 2 = -\frac{1}{2}x + 5 \]
Solve for \( x \):
Multiply through by 6 (the least common multiple of the denominators):
\[ 6\left(\frac{2}{3}x\right) - 6(2) = 6\left(-\frac{1}{2}x\right) + 6(5) \]
This simplifies to:
\[ 4x - 12 = -3x + 30 \]
Add \( 3x \) to both sides:
\[ 4x + 3x - 12 = 30 \]
Combine:
\[ 7x - 12 = 30 \]
Add 12 to both sides:
\[ 7x = 42 \]
Divide by 7:
\[ x = 6 \]
Step 3: Substitute for \( y \):
Plug \( x = 6 \) back into either equation (using the red line):
\[ y = -\frac{1}{2}(6) + 5 = -3 + 5 = 2 \]
Solution of the System of Equations:
The solution to the system, or the intersection point, is \( (6, 2) \).
Sketching the Graph:
- The blue line passes through (0, -2), (3, 0), and (6, 2).
- The red line passes through (0, 5), (6, 2).
- Both lines intersect at (6, 2).
Final Summary:
Intersection Point: \( (6, 2) \)
You can now take these points to graph the lines on your graph paper, ensuring your graph's y-values do not exceed 8.
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