Question

Solve the following system by the substitution method. Check the​ solution(s).


3 x plus 5 y
equals9
xminus2y
equals3

Answer 1

To solve the system of equations using the substitution method, we have the following two equations:

1) \( 3x + 5y = 9 \)
2) \( x - 2y = 3 \)

First, let's solve the second equation for \(x\):

\[
x = 2y + 3
\]

Now substitute this expression for \(x\) into the first equation:

\[
3(2y + 3) + 5y = 9
\]

Distribute the 3:

\[
6y + 9 + 5y = 9
\]

Combine like terms:

\[
11y + 9 = 9
\]

Subtract 9 from both sides:

\[
11y = 0
\]

Now, divide by 11:

\[
y = 0
\]

Now that we have \(y\), we can substitute it back into the expression we found for \(x\):

\[
x = 2(0) + 3 = 3
\]

So the solution to the system is:

\[
x = 3, \quad y = 0
\]

Now, let's check the solution by substituting \(x\) and \(y\) back into the original equations:

1) Check \(3x + 5y = 9\):

\[
3(3) + 5(0) = 9 \rightarrow 9 = 9 \quad \text{(True)}
\]

2) Check \(x - 2y = 3\):

\[
3 - 2(0) = 3 \rightarrow 3 = 3 \quad \text{(True)}
\]

Both equations hold true, so the solution \( (x, y) = (3, 0) \) is correct.

Thus, the final solution is:

\[
\boxed{(3, 0)}
\]