Answer :
To solve the system of equations by graphing, we should first rewrite each equation in slope-intercept form ( \(y = mx + b\) ), which makes them easier to graph. The given system of equations is:
[tex]\[ \begin{aligned} -2x - 2y &= 20 \\ 2x - y &= -8 \end{aligned} \][/tex]
1. Start with the first equation:
[tex]\[ -2x - 2y = 20 \][/tex]
Divide every term by \(-2\) to simplify:
[tex]\[ x + y = -10 \][/tex]
Next, solve for \( y \):
[tex]\[ y = -x - 10 \][/tex]
2. Now, take the second equation:
[tex]\[ 2x - y = -8 \][/tex]
Solve for \( y \) by isolating it on one side:
[tex]\[ y = 2x + 8 \][/tex]
Now we have the two equations in slope-intercept form:
[tex]\[ \begin{aligned} y &= -x - 10 \\ y &= 2x + 8 \end{aligned} \][/tex]
3. We can now graph these equations.
For the first equation, \(y = -x - 10\):
- The y-intercept is \(-10\) (the point where the line crosses the y-axis), so plot the point \((0, -10)\).
- The slope is \(-1\), indicating that for every unit increase in \(x\), \(y\) decreases by 1. Starting from \((0, -10)\), moving one unit to the right gives \((1, -11)\).
For the second equation, \(y = 2x + 8\):
- The y-intercept is \(8\), so plot the point \((0, 8)\).
- The slope is \(2\), indicating that \(y\) increases by 2 for every 1 unit increase in \(x\). Starting from \((0, 8)\), moving one unit to the right gives \((1, 10)\).
4. Connect the points for each line to extend them and identify their intersection point on the graph.
Checking the intersection carefully on the graph, you will find that the two lines intersect at the point \((-6, -4)\).
So, the solution to the system of equations is:
[tex]\[ (x, y) = (-6, -4) \][/tex]
[tex]\[ \begin{aligned} -2x - 2y &= 20 \\ 2x - y &= -8 \end{aligned} \][/tex]
1. Start with the first equation:
[tex]\[ -2x - 2y = 20 \][/tex]
Divide every term by \(-2\) to simplify:
[tex]\[ x + y = -10 \][/tex]
Next, solve for \( y \):
[tex]\[ y = -x - 10 \][/tex]
2. Now, take the second equation:
[tex]\[ 2x - y = -8 \][/tex]
Solve for \( y \) by isolating it on one side:
[tex]\[ y = 2x + 8 \][/tex]
Now we have the two equations in slope-intercept form:
[tex]\[ \begin{aligned} y &= -x - 10 \\ y &= 2x + 8 \end{aligned} \][/tex]
3. We can now graph these equations.
For the first equation, \(y = -x - 10\):
- The y-intercept is \(-10\) (the point where the line crosses the y-axis), so plot the point \((0, -10)\).
- The slope is \(-1\), indicating that for every unit increase in \(x\), \(y\) decreases by 1. Starting from \((0, -10)\), moving one unit to the right gives \((1, -11)\).
For the second equation, \(y = 2x + 8\):
- The y-intercept is \(8\), so plot the point \((0, 8)\).
- The slope is \(2\), indicating that \(y\) increases by 2 for every 1 unit increase in \(x\). Starting from \((0, 8)\), moving one unit to the right gives \((1, 10)\).
4. Connect the points for each line to extend them and identify their intersection point on the graph.
Checking the intersection carefully on the graph, you will find that the two lines intersect at the point \((-6, -4)\).
So, the solution to the system of equations is:
[tex]\[ (x, y) = (-6, -4) \][/tex]
