To solve the system of equations graphically, we will plot the two equations:
- \( y = -x - 2 \)
- \( y = \frac{1}{6}x + 5 \)
Step 1: Plot the first equation
For the first equation \( y = -x - 2 \):
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Find the y-intercept: When \( x = 0 \), \[ y = -0 - 2 = -2 \] So the point is (0, -2).
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Find another point: When \( x = 2 \), \[ y = -2 - 2 = -4 \] So the point is (2, -4).
Step 2: Plot the second equation
For the second equation \( y = \frac{1}{6}x + 5 \):
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Find the y-intercept: When \( x = 0 \), \[ y = \frac{1}{6}(0) + 5 = 5 \] So the point is (0, 5).
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Find another point: When \( x = 6 \), \[ y = \frac{1}{6}(6) + 5 = 1 + 5 = 6 \] So the point is (6, 6).
Step 3: Graphing the lines
Now you can plot the points you've found:
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For \( y = -x - 2 \):
- Plot (0, -2) and (2, -4) and draw the line through these points.
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For \( y = \frac{1}{6}x + 5 \):
- Plot (0, 5) and (6, 6) and draw the line through these points.
Step 4: Finding the Intersection
The solution to the system of equations is the point where the two lines intersect. From the graph, identify the point where the two lines cross.
Step 5: Analyzing the Graph
You will likely find that the lines intersect at a certain point (let's say \( (x, y) \)). You can also compute the intersection analytically by setting the two equations equal to each other:
\[ -x - 2 = \frac{1}{6}x + 5 \]
Combine like terms and solve for \( x \):
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Multiply through by 6 to eliminate the fraction: \[ -6x - 12 = x + 30 \] \[ -7x = 42 \] \[ x = -6 \]
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Substitute \( x = -6 \) back into one of the original equations to find \( y \): \[ y = -(-6) - 2 = 6 - 2 = 4 \]
So the solution to the system of equations is \( (-6, 4) \).
You can confirm this graphically by ensuring the lines intersect at this point on your plotted graph.
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