To solve the system of equations using substitution, we have the following two equations:
- \( 2x + 3y = 11 \) (Equation 1)
- \( y = 3x \) (Equation 2)
Step 1: Substitute Equation 2 into Equation 1
Since Equation 2 expresses \( y \) in terms of \( x \), we can substitute \( y \) with \( 3x \) in Equation 1.
\[ 2x + 3(3x) = 11 \]
Step 2: Simplify the equation
Now, simplify the equation:
\[ 2x + 9x = 11 \]
Combine like terms:
\[ 11x = 11 \]
Step 3: Solve for \( x \)
Now, divide both sides by 11:
\[ x = 1 \]
Step 4: Substitute \( x \) back to find \( y \)
Now that we have \( x \), we will substitute \( x = 1 \) back into Equation 2 to find \( y \):
\[ y = 3(1) = 3 \]
Step 5: Write the solution
Thus, the solution to the system of equations is:
\[ (x, y) = (1, 3) \]
Step 6: Verify the solution
We can substitute these values back into the original equations to verify that both equations are satisfied:
- Check Equation 1:
\[ 2(1) + 3(3) = 2 + 9 = 11 \quad \text{(True)} \]
- Check Equation 2:
\[ y = 3x \implies 3 = 3(1) \quad \text{(True)} \]
Both equations are satisfied, so our solution \( (1, 3) \) is correct.
Final Answer:
The solution to the system of equations is:
\[ (x, y) = (1, 3) \]