To find the solution to the system of equations given by \( y = \frac{5}{8}x - 3 \) and \( y = -\frac{3}{4}x + 8 \), we start by plotting both equations on a graph and then find their intersection point, which gives us the solution to the system.
Step 1: Graph the equations
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Equation 1: \( y = \frac{5}{8}x - 3 \)
- The slope is \( \frac{5}{8} \) and the y-intercept is -3.
- To plot it, start from the y-intercept (0, -3).
- Then use the slope to find another point. For every 8 units you move to the right (increase in x), move up 5 units (increase in y).
- Points:
- (0, -3)
- (8, \frac{5}{8} \cdot 8 - 3) = (8, 5 - 3) = (8, 2)
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Equation 2: \( y = -\frac{3}{4}x + 8 \)
- The slope is -\(\frac{3}{4}\) and the y-intercept is 8.
- Start at (0, 8).
- For this slope, for every 4 units you move to the right, move down 3 units.
- Points:
- (0, 8)
- (4, -\frac{3}{4} \cdot 4 + 8) = (4, -3 + 8) = (4, 5)
Step 2: Intersection point
To find the intersection of the two lines, set the equations equal to each other:
\[ \frac{5}{8}x - 3 = -\frac{3}{4}x + 8 \]
Step 3: Solve for \(x\)
- Multiply through by 8 to eliminate the fractions:
\[ 5x - 24 = -6x + 64 \]
- Combine like terms:
\[ 5x + 6x = 64 + 24 \] \[ 11x = 88 \]
- Divide by 11:
\[ x = 8 \]
Step 4: Find \(y\)
Now substitute \(x = 8\) into either of the original equations to find \(y\). Using the first equation:
\[ y = \frac{5}{8}(8) - 3 = 5 - 3 = 2 \]
Conclusion
The solution to the system of equations is:
\[ \boxed{(8, 2)} \]
This means that the two lines intersect at the point \((8, 2)\).