To solve the system of equations using substitution, we start with the two equations provided:
- \( y = 3x - 2 \) (Equation 1)
- \( 2x + 4y = 20 \) (Equation 2)
Step 1: Substitute Equation 1 into Equation 2
Since Equation 1 gives us \( y \) in terms of \( x \), we can substitute this expression for \( y \) into Equation 2.
Substituting \( y = 3x - 2 \) into Equation 2:
\[ 2x + 4(3x - 2) = 20 \]
Step 2: Simplify the Equation
Now we expand and simplify the equation:
\[ 2x + 12x - 8 = 20 \]
Combine like terms:
\[ 14x - 8 = 20 \]
Step 3: Solve for \( x \)
Next, we isolate \( x \). First, add 8 to both sides:
\[ 14x = 20 + 8 \] \[ 14x = 28 \]
Now, divide both sides by 14:
\[ x = \frac{28}{14} = 2 \]
Step 4: Substitute \( x \) back into Equation 1 to find \( y \)
Now that we have \( x = 2 \), we will substitute this value back into Equation 1 to find \( y \):
\[ y = 3(2) - 2 \] \[ y = 6 - 2 = 4 \]
Step 5: Write the solution
The solution to the system of equations is: \[ (x, y) = (2, 4) \]
Step 6: Verify the solution
Finally, let's verify that this solution satisfies both original equations.
- Check Equation 1:
\[ y = 3(2) - 2 = 6 - 2 = 4 \quad \text{(True)} \]
- Check Equation 2:
\[ 2(2) + 4(4) = 4 + 16 = 20 \quad \text{(True)} \]
Since both equations are satisfied by \( (2, 4) \), we conclude that the solution is correct.
Thus, the solution to the system of equations is: \[ \boxed{(2, 4)} \]