Question

Find the solution to the problems:
6x+3y=33
5x-8y=-25

Answer 1

To solve the system of equations:

1. \( 6x + 3y = 33 \) (Equation 1)
2. \( 5x - 8y = -25 \) (Equation 2)

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

First, let's simplify Equation 1 by dividing everything by 3:

\[
2x + y = 11 \quad (Equation \; 1')
\]

Now, we can express \( y \) in terms of \( x \):

\[
y = 11 - 2x \quad (Equation \; 3)
\]

Now, we can substitute this expression for \( y \) into Equation 2:

\[
5x - 8(11 - 2x) = -25
\]
\[
5x - 88 + 16x = -25
\]
\[
21x - 88 = -25
\]
\[
21x = -25 + 88
\]
\[
21x = 63
\]
\[
x = 3
\]

Now that we have \( x \), we can substitute it back into Equation 3 to find \( y \):

\[
y = 11 - 2(3)
\]
\[
y = 11 - 6
\]
\[
y = 5
\]

The solution to the system of equations is:

\[
(x, y) = (3, 5)
\]

To verify, we can plug \( x = 3 \) and \( y = 5 \) back into the original equations:

1. For Equation 1:
\[
6(3) + 3(5) = 18 + 15 = 33 \quad \text{(True)}
\]

2. For Equation 2:
\[
5(3) - 8(5) = 15 - 40 = -25 \quad \text{(True)}
\]

Both equations are satisfied, confirming our solution. Thus, the final answer is:

\[
\boxed{(3, 5)}
\]