To solve the system of equations graphically, we need to graph the equations and find their intersection point.
The equations given are:
- \(y = -2x - 1\)
- \(3x - y = -7\)
First, let's rewrite the second equation in slope-intercept form (y = mx + b):
\[ 3x - y = -7 \ -y = -3x - 7 \ y = 3x + 7 \]
Now we have the following two equations to graph:
- \(y = -2x - 1\)
- \(y = 3x + 7\)
Steps to Graph
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Graph the first equation: \(y = -2x - 1\)
- The y-intercept is at (0, -1).
- The slope is -2, meaning that for every 1 unit you move to the right (increase x by 1), you move down 2 units (decrease y by 2).
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Graph the second equation: \(y = 3x + 7\)
- The y-intercept is at (0, 7).
- The slope is 3, meaning that for every 1 unit you move to the right, you move up 3 units.
Finding the Intersection
Using a graphing calculator or by hand, you can plot both lines. The intersection point is where both equations are true at the same time.
From the options provided, you would need to check which point, when plugged into both equations, is valid.
Check the options:
-
For (-1.6, 2.2):
- For \(y = -2x - 1\): \(2.2 = -2(-1.6) - 1 \rightarrow 2.2 = 3.2 - 1 \rightarrow 2.2 = 2.2\) (True)
- For \(y = 3x + 7\): \(2.2 = 3(-1.6) + 7 \rightarrow 2.2 = -4.8 + 7 \rightarrow 2.2 = 2.2\) (True)
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For (-1.5, 0):
- For \(y = -2(-1.5) - 1 \Rightarrow 0 = 3 - 1 \Rightarrow 0 = 2\) (False)
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For (-1, -1):
- \(y = -2(-1) - 1 \Rightarrow -1 = 2 - 1 \Rightarrow -1 = 1\) (False)
-
For (0, 2):
- \(y = -2(0) - 1 \Rightarrow 2 = -1\) (False)
Conclusion
The valid solution is (-1.6, 2.2), which satisfies both equations.
Therefore, the solution to the system of equations is:
Solution is (-1.6, 2.2)