Solve the System of Equations: 4x + 3y = -11 and 3x - 2y = -4

To solve this problem, we'll follow these steps:

• Step 1: Align the coefficients to eliminate one variable by manipulating the given equations.
• Step 2: Solve for one variable using the elimination method.
• Step 3: Substitute back to find the other variable.
• Step 4: Verify the solution by substituting back into the original equations.

Let's work through each step:

Step 1: Multiply the first equation by 2 and the second equation by 3 to align the coefficients of $y$.

This gives us:

$8x + 6y = -22$ (Equation 1 multiplied by 2)
$9x - 6y = -12$ (Equation 2 multiplied by 3)

Step 2: Add the two equations to eliminate $y$.

$(8x + 6y) + (9x - 6y) = -22 - 12$
$17x = -34$

Solve for $x$:

$x = \frac{-34}{17} = -2$

Step 3: Substitute $x = -2$ back into one of the original equations to solve for $y$. Using the first equation:

$4(-2) + 3y = -11$
$-8 + 3y = -11$
$3y = -11 + 8$
$3y = -3$
$y = \frac{-3}{3} = -1$

Step 4: Verify the solution by substituting $x = -2$ and $y = -1$ into the second original equation:

$3(-2) - 2(-1) = -4$
$-6 + 2 = -4$
$-4 = -4$ which holds true.

Therefore, the solution to the system of equations is $\mathbf{(x = -2, y = -1)}$.