To solve the system of equations by graphing, we'll first rewrite the two equations and then plot them on a coordinate plane.
The equations are:
- \( y = 2x + 4 \)
- \( y = -x - 8 \)
Step 1: Graph the first equation \(y = 2x + 4\)
This equation is in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
- Y-intercept: When \(x = 0\), \(y = 4\). So the point (0, 4) is on the graph.
- Slope: The slope \(m = 2\) means that for every 1 unit increase in \(x\), \(y\) increases by 2 units. Starting from (0, 4):
- When \(x = 1\): \(y = 2(1) + 4 = 6\) → point (1, 6)
- When \(x = -1\): \(y = 2(-1) + 4 = 2\) → point (-1, 2)
Step 2: Graph the second equation \(y = -x - 8\)
The slope-intercept form is also applicable here.
- Y-intercept: When \(x = 0\), \(y = -8\). So the point (0, -8) is on the graph.
- Slope: The slope \(m = -1\) means that for every 1 unit increase in \(x\), \(y\) decreases by 1 unit. Starting from (0, -8):
- When \(x = 1\): \(y = -1 - 8 = -9\) → point (1, -9)
- When \(x = -1\): \(y = 1 - 8 = -7\) → point (-1, -7)
Step 3: Plot the points and draw the lines
Now, we can plot the following points:
For \(y = 2x + 4\):
- (0, 4)
- (1, 6)
- (-1, 2)
For \(y = -x - 8\):
- (0, -8)
- (1, -9)
- (-1, -7)
Once you have plotted these points on the graph, draw the lines for each equation.
Step 4: Identify the intersection point
The solution to the system of equations is the point where the two lines intersect.
By examining the graphs, you will see that the lines intersect at the point \((-6, -8)\).
Conclusion
The solution to the system of equations: \[ \begin{align*} y &= 2x + 4 \ y &= -x - 8 \end{align*} \] is \(x = -6\) and \(y = -8\) or the point \((-6, -8)\).
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