Bot GPT 4-o mini To solve the system of equations given by \( y = -\frac{1}{4}x + 8 \) and \( y = \frac{1}{2}x + 2 \), we can first graph the two lines and then find the point of intersection.
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Graphing the equations:
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For \( y = -\frac{1}{4}x + 8 \):
- The y-intercept is \( 8 \). This means the line crosses the y-axis at \( (0, 8) \).
- To find another point, you can choose \( x = 4 \): \[ y = -\frac{1}{4}(4) + 8 = -1 + 8 = 7 \text{, thus the point (4, 7)} \]
- So we have two points: \( (0, 8) \) and \( (4, 7) \).
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For \( y = \frac{1}{2}x + 2 \):
- The y-intercept is \( 2 \). This means this line crosses the y-axis at \( (0, 2) \).
- To find another point, you can choose \( x = 4 \): \[ y = \frac{1}{2}(4) + 2 = 2 + 2 = 4 \text{, thus the point (4, 4)} \]
- So we have two points: \( (0, 2) \) and \( (4, 4) \).
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Finding where the lines intersect:
To find the solution algebraically, set the equations equal to each other:
\[ -\frac{1}{4}x + 8 = \frac{1}{2}x + 2 \]
Now, eliminate the fractions by multiplying the entire equation by \( 4 \):
\[ -x + 32 = 2x + 8 \]
Rearranging gives:
\[ 32 - 8 = 2x + x \]
\[ 24 = 3x \]
So:
\[ x = 8 \]
Now substitute \( x = 8 \) back into either original equation to find \( y \):
Using \( y = \frac{1}{2}x + 2 \):
\[ y = \frac{1}{2}(8) + 2 = 4 + 2 = 6 \]
Thus, the solution to the system of equations is:
\[ \boxed{(8, 6)} \]
This indicates that the lines intersect at the point \( (8, 6) \).
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