Question

solve the system of equations
-3x-4y=-23
3x-5y=5

Answer 1

To solve the system of equations

\[
-3x - 4y = -23 \quad \text{(1)}
\]

\[
3x - 5y = 5 \quad \text{(2)}
\]

we can use the method of elimination or substitution. Here, I will use elimination.

First, let's eliminate \( x \). We can do this by manipulating both equations. Let's start by multiplying equation (1) by \( -1 \) to get the coefficients of \( x \) to be the same:

\[
3x + 4y = 23 \quad \text{(3)}
\]

Now we can add equation (2) and equation (3):

\[
(3x + 4y) + (3x - 5y) = 23 + 5
\]

This simplifies to:

\[
6x - y = 28 \quad \text{(4)}
\]

Now, let's isolate \( y \) in equation (4):

\[
-y = 28 - 6x
\]

\[
y = 6x - 28 \quad \text{(5)}
\]

Now, we can substitute equation (5) back into either original equation for \( x \). We'll use equation (2):

\[
3x - 5(6x - 28) = 5
\]

Expanding the equation gives:

\[
3x - 30x + 140 = 5
\]

Combining like terms:

\[
-27x + 140 = 5
\]

Now, isolating \( x \):

\[
-27x = 5 - 140
\]

\[
-27x = -135
\]

\[
x = \frac{-135}{-27} = 5
\]

Now that we have \( x \), we can substitute back into equation (5) to find \( y \):

\[
y = 6(5) - 28
\]

\[
y = 30 - 28
\]

\[
y = 2
\]

Thus, the solution to the system of equations is

\[
(x, y) = (5, 2).
\]

To verify, we can substitute \( x = 5 \) and \( y = 2 \) back into the original equations:

1. \( -3(5) - 4(2) = -15 - 8 = -23 \) (True)
2. \( 3(5) - 5(2) = 15 - 10 = 5 \) (True)

Both equations are satisfied, confirming our solution is correct:

\[
\boxed{(5, 2)}.
\]