Solve the system of equations. 5x-4y=-44, -x+3y=22

Answer 1

To solve the system of linear equations:

\[ \begin{cases}
5x - 4y = -44 \\
-x + 3y = 22
\end{cases} \]

we can use either the substitution method or the elimination method. Here, we'll use the elimination method.

1. First, we'll multiply the second equation by 5 to align the coefficients of \( x \):

\[ -x + 3y = 22 \implies -5x + 15y = 110 \]

So the system becomes:

\[ \begin{cases}
5x - 4y = -44 \\
-5x + 15y = 110
\end{cases} \]

2. Now, add the two equations together to eliminate \( x \):

\[ (5x - 4y) + (-5x + 15y) = -44 + 110 \]

This simplifies to:

\[ (5x - 5x) + (-4y + 15y) = 66 \]

\[ 0 + 11y = 66 \]

\[ 11y = 66 \]

3. Solve for \( y \):

\[ y = \frac{66}{11} \]

\[ y = 6 \]

4. Substitute \( y = 6 \) into one of the original equations to solve for \( x \). Let's use the second equation:

\[ -x + 3y = 22 \]

\[ -x + 3(6) = 22 \]

\[ -x + 18 = 22 \]

\[ -x = 22 - 18 \]

\[ -x = 4 \]

\[ x = -4 \]

So, the solution to the system of equations is:

\[ x = -4, \quad y = 6 \]

To verify, substitute \( x = -4 \) and \( y = 6 \) back into the original equations:

1. Check the first equation:

\[ 5(-4) - 4(6) = -20 - 24 = -44 \] (which is correct)

2. Check the second equation:

\[ -(-4) + 3(6) = 4 + 18 = 22 \] (which is correct)

Therefore, the solution \((x, y) = (-4, 6)\) is correct.