Certainly! Let's solve the given system of equations by graphing and find the point of intersection.
The system of equations is:
[tex]\[
\left\{\begin{array}{l}
-2x - y = -12 \\
2x - 3y = 4
\end{array}\right.
\][/tex]
### Step 1: Rewrite each equation in slope-intercept form ( [tex]\( y = mx + b \)[/tex] ).
First Equation:
[tex]\[
-2x - y = -12
\][/tex]
Solve for [tex]\( y \)[/tex]:
[tex]\[
- y = 2x - 12
\][/tex]
[tex]\[
y = -2x + 12
\][/tex]
So, the first equation in slope-intercept form is:
[tex]\[
y = -2x + 12
\][/tex]
Second Equation:
[tex]\[
2x - 3y = 4
\][/tex]
Solve for [tex]\( y \)[/tex]:
[tex]\[
2x - 4 = 3y
\][/tex]
[tex]\[
3y = 2x - 4
\][/tex]
[tex]\[
y = \frac{2}{3}x - \frac{4}{3}
\][/tex]
So, the second equation in slope-intercept form is:
[tex]\[
y = \frac{2}{3}x - \frac{4}{3}
\][/tex]
### Step 2: Graph each equation on the same coordinate plane.
First Equation [tex]\( y = -2x + 12 \)[/tex]:
- The y-intercept is at [tex]\( (0, 12) \)[/tex].
- The slope is [tex]\(-2\)[/tex], which means for every 1 unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 2 units.
Second Equation [tex]\( y = \frac{2}{3}x - \frac{4}{3} \)[/tex]:
- The y-intercept is at [tex]\( \left(0, -\frac{4}{3}\right) \)[/tex].
- The slope is [tex]\(\frac{2}{3}\)[/tex], which means for every 3 units increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 2 units.
### Step 3: Identify the point of intersection.
Plotting both lines on the graph and finding their intersection:
- The first line [tex]\( y = -2x + 12 \)[/tex] intercepts the y-axis at [tex]\( (0, 12) \)[/tex] and descends with a slope of [tex]\(-2\)[/tex].
- The second line [tex]\( y = \frac{2}{3}x - \frac{4}{3} \)[/tex] intercepts the y-axis at [tex]\( \left(0, -\frac{4}{3}\right) \)[/tex] and ascends with a slope of [tex]\(\frac{2}{3}\)[/tex].
After drawing these lines, the point where they intersect is the solution to the system of equations.
From the given solution, the point of intersection is:
[tex]\[
(x, y) = (5, 2)
\][/tex]
### Verification:
Let's verify by substituting [tex]\( (5, 2) \)[/tex] into the original equations.
First Equation:
[tex]\[
-2(5) - 2 = -12
\][/tex]
[tex]\[
-10 - 2 = -12
\][/tex]
True.
Second Equation:
[tex]\[
2(5) - 3(2) = 4
\][/tex]
[tex]\[
10 - 6 = 4
\][/tex]
True.
Thus, the point [tex]\( (5, 2) \)[/tex] satisfies both equations, confirming it as the point of intersection.
### Conclusion:
The point of intersection for the given system of equations is:
[tex]\[
\boxed{(5, 2)}
\][/tex]
