To solve the system of equations:
[tex]\[
\begin{cases}
2x - 3y = -18 \\
3x + y = -5
\end{cases}
\][/tex]
We will use the method of solving systems of linear equations, producing a step-by-step solution.
### Step 1: Write down the equations.
The system of equations is:
1. [tex]\(2x - 3y = -18\)[/tex]
2. [tex]\(3x + y = -5\)[/tex]
### Step 2: Solve one of the equations for one variable.
Let's solve the second equation for [tex]\(y\)[/tex]:
[tex]\[3x + y = -5\][/tex]
[tex]\[y = -5 - 3x\][/tex]
### Step 3: Substitute this expression into the first equation.
Substituting [tex]\(y = -5 - 3x\)[/tex] into the first equation:
[tex]\[2x - 3(-5 - 3x) = -18\][/tex]
### Step 4: Simplify and solve for [tex]\(x\)[/tex].
Distribute the [tex]\(-3\)[/tex]:
[tex]\[2x + 15 + 9x = -18\][/tex]
Combine like terms:
[tex]\[11x + 15 = -18\][/tex]
Subtract 15 from both sides:
[tex]\[11x = -18 - 15\][/tex]
[tex]\[11x = -33\][/tex]
Divide by 11:
[tex]\[x = -3\][/tex]
### Step 5: Substitute [tex]\(x = -3\)[/tex] back into the equation solved for [tex]\(y\)[/tex].
Using [tex]\(y = -5 - 3x\)[/tex]:
[tex]\[y = -5 - 3(-3)\][/tex]
[tex]\[y = -5 + 9\][/tex]
[tex]\[y = 4\][/tex]
### Conclusion: The solution to the system of equations is [tex]\(x = -3\)[/tex] and [tex]\(y = 4\)[/tex].
To visualize this solution on the graph:
1. Plot the point [tex]\((-3, 4)\)[/tex] on the Cartesian plane.
2. This point represents the intersection of the lines representing the two equations.
This intersection point [tex]\((-3, 4)\)[/tex] confirms that the solution to the system of equations is [tex]\((x, y) = (-3, 4)\)[/tex].
