Solve the following system by graphing and identify the point of intersection.

[tex]\[
\left\{
\begin{array}{l}
-2x - y = -12 \\
2x - 3y = 4
\end{array}
\right.
\][/tex]



Answer :

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]