To solve the system of equations graphically, we first need to rewrite the given equations in slope-intercept form (y = mx + b) to identify their slopes and intercepts.
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First equation: \( y = x - 4 \)
This is already in slope-intercept form, where the slope \( m = 1 \) and the y-intercept \( b = -4 \).
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Second equation: \( x + 2y = 10 \)
To convert this to slope-intercept form, solve for \( y \):
\[ 2y = -x + 10 \] \[ y = -\frac{1}{2}x + 5 \]
This gives us a slope of \( m = -\frac{1}{2} \) and a y-intercept of \( b = 5 \).
Now we can plot both equations on a graph:
For the first equation, \( y = x - 4 \):
- When \( x = 0 \): \( y = 0 - 4 = -4 \) (point: (0, -4))
- When \( x = 4 \): \( y = 4 - 4 = 0 \) (point: (4, 0))
- When \( x = 6 \): \( y = 6 - 4 = 2 \) (point: (6, 2))
For the second equation, \( y = -\frac{1}{2}x + 5 \):
- When \( x = 0 \): \( y = -\frac{1}{2}(0) + 5 = 5 \) (point: (0, 5))
- When \( x = 10 \): \( y = -\frac{1}{2}(10) + 5 = 0 \) (point: (10, 0))
- When \( x = 2 \): \( y = -\frac{1}{2}(2) + 5 = 4 \) (point: (2, 4))
Now, you can plot these points on a graph.
After plotting, you can draw the lines for both equations. The point where the two lines intersect is the solution to the system of equations.
Finding the intersection algebraically: We can also solve it algebraically to confirm the graphical solution:
- From the first equation: \( y = x - 4 \)
- Substitute \( y \) in the second equation: \[ x + 2(x - 4) = 10 \] \[ x + 2x - 8 = 10 \] \[ 3x - 8 = 10 \] \[ 3x = 18 \] \[ x = 6 \]
Now substitute \( x = 6 \) back into the first equation to find \( y \): \[ y = 6 - 4 = 2 \]
Thus, the intersection point, which is the solution to the system of equations, is \((6, 2)\).
Final Solution: The solution to the system of equations is \( (6, 2) \).
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