To find the solution to the given system of equations, we can start by rewriting them for clarity.
- \( y = \frac{1}{2} x + 2 \)
- \( y = -\frac{1}{4} x + 8 \)
Step 1: Graph the equations
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Graphing \( y = \frac{1}{2} x + 2 \):
- The y-intercept is 2 (when \( x = 0 \), \( y = 2 \)).
- The slope is \( \frac{1}{2} \), meaning for every increase of 1 in \( x \), \( y \) increases by \( \frac{1}{2} \).
- Plot points:
- For \( x = 0 \): \( (0, 2) \)
- For \( x = 2 \): \( y = \frac{1}{2}(2) + 2 = 3 \) (point \( (2, 3) \))
- For \( x = 4 \): \( y = \frac{1}{2}(4) + 2 = 4 \) (point \( (4, 4) \))
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Graphing \( y = -\frac{1}{4} x + 8 \):
- The y-intercept is 8 (when \( x = 0 \), \( y = 8 \)).
- The slope is \(-\frac{1}{4}\), meaning for every increase of 1 in \( x \), \( y \) decreases by \( \frac{1}{4} \).
- Plot points:
- For \( x = 0 \): \( (0, 8) \)
- For \( x = 4 \): \( y = -\frac{1}{4}(4) + 8 = 7 \) (point \( (4, 7) \))
- For \( x = 8 \): \( y = -\frac{1}{4}(8) + 8 = 6 \) (point \( (8, 6) \))
Step 2: Find the intersection
Set the right-hand sides of the equations equal to each other to find the intersection point:
\[ \frac{1}{2} x + 2 = -\frac{1}{4} x + 8 \]
Step 3: Solve for \( x \)
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Multiply the entire equation by 4 to eliminate the fractions:
\[ 4\left(\frac{1}{2} x + 2\right) = 4\left(-\frac{1}{4} x + 8\right) \]
This simplifies to:
\[ 2x + 8 = -x + 32 \]
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Rearranging gives:
\[ 2x + x = 32 - 8 \] \[ 3x = 24 \] \[ x = 8 \]
Step 4: Substitute \( x \) back to find \( y \)
Using the first equation:
\[ y = \frac{1}{2}(8) + 2 = 4 + 2 = 6 \]
The point of intersection, which is the solution to the system of equations, is:
\[ (8, 6) \]
Conclusion:
The solution to the system of equations is the point \( (8, 6) \).