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

Question

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:

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