To solve the system of equations graphically, we need to first rewrite both equations in a suitable forms:
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The first equation is given as: \[ y = \frac{5}{2}x - 6 \] This is already in slope-intercept form \(y = mx + b\), where \(m = \frac{5}{2}\) (the slope), and \(b = -6\) (the y-intercept).
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The second equation is: \[ x + y = 8 \] We can rearrange it to slope-intercept form: \[ y = -x + 8 \] Here, the slope \(m = -1\) and the y-intercept \(b = 8\).
Steps to Graph the Equations
Graph the first equation \(y = \frac{5}{2}x - 6\):
- Start by plotting the y-intercept \((0, -6)\).
- Use the slope \(\frac{5}{2}\). From the y-intercept, move up 5 units (rise) and right 2 units (run) to find another point, which would be \((2, -1)\).
- Draw the line through these points.
Graph the second equation \(y = -x + 8\):
- Plot the y-intercept \((0, 8)\).
- Use the slope \(-1\). From \((0, 8)\), move down 1 unit and right 1 unit to find another point, which would be \((1, 7)\).
- Draw the line through these points.
Finding the Intersection Point
After plotting both lines on the same set of axes, the intersection point of the two lines represents the solution to the system of equations.
To find the intersection, we can also set the equations equal to one another:
From the two equations: \[ \frac{5}{2}x - 6 = -x + 8 \]
To solve for \(x\):
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Add \(x\) to both sides: \[ \frac{5}{2}x + x - 6 = 8 \ \frac{5}{2}x + \frac{2}{2}x = 8 + 6 \ \frac{7}{2}x = 14 \]
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Multiply both sides by \(\frac{2}{7}\): \[ x = 14 \cdot \frac{2}{7} = 4 \]
Now substitute \(x = 4\) back into one of the original equations to find \(y\): \[ y = \frac{5}{2}(4) - 6 = 10 - 6 = 4 \]
Therefore, the intersection point, and thus the solution to the system of equations is: \[ \boxed{(4, 4)} \]
This point can be marked on the graph as the solution to the system.