Question

solve the system of equations $-2x - 2y = -24$ and $-3x - 2y = -33$ by combining the equations.\\(\

$$\begin{pmatrix}-2x&-2y&=-24\\\\-3x&-2y&=-33\\end{pmatrix}$$

\\)\\(\

$$\begin{align*}-2x&-2y&=&-24\\\\-3x&-2y&=&-33\\\\\\hline 0x&+0y&=&\\end{align*}$$

\\)

Explanation:

Step1: Subtract the two equations

To eliminate \( y \), we subtract the second equation from the first equation (or vice versa). Let's subtract the second equation \(-3x - 2y = -33\) from the first equation \(-2x - 2y = -24\).

\[

$$\begin{align*} (-2x - 2y) - (-3x - 2y) &= -24 - (-33)\\ -2x - 2y + 3x + 2y &= -24 + 33\\ x + 0y &= 9 \end{align*}$$

\]

Step2: Solve for \( x \)

From the result of Step 1, we have \( x = 9 \).

Step3: Substitute \( x = 9 \) into one of the original equations to find \( y \)

Let's use the first equation \(-2x - 2y = -24\). Substitute \( x = 9 \):

\[

$$\begin{align*} -2(9) - 2y &= -24\\ -18 - 2y &= -24\\ -2y &= -24 + 18\\ -2y &= -6\\ y &= 3 \end{align*}$$

\]

Answer:

The solution to the system of equations is \( x = 9 \) and \( y = 3 \). The combined equation gives \( 1x + 0y = 9 \) (or simply \( x = 9 \)).