To solve the system of equations using the elimination method, we'll start with the given equations:
-
\( 2x + 5y = 20 \) \quad (Equation 1)
-
\( 3x - 5y = 5 \) \quad (Equation 2)
Step 1: Align the equations
We will align the equations for clarity.
\[ \begin{align*}
- & \quad 2x + 5y = 20 \
- & \quad 3x - 5y = 5 \ \end{align*} \]
Step 2: Add the equations
To eliminate \( y \), we can add the two equations, as they have coefficients of \( +5y \) and \( -5y \).
First, we can just add the two equations directly:
\[ (2x + 5y) + (3x - 5y) = 20 + 5 \]
This simplifies to:
\[ 5x + 0y = 25 \]
Thus, we have:
\[ 5x = 25 \]
Step 3: Solve for \( x \)
Now, divide both sides by \( 5 \):
\[ x = \frac{25}{5} = 5 \]
Step 4: Substitute \( x \) back into one of the original equations
We can substitute \( x = 5 \) into Equation 1 to solve for \( y \):
\[ 2(5) + 5y = 20 \]
This simplifies to:
\[ 10 + 5y = 20 \]
Step 5: Solve for \( y \)
Now, isolate \( y \) by subtracting \( 10 \) from both sides:
\[ 5y = 20 - 10 \]
\[ 5y = 10 \]
Now, divide by \( 5 \):
\[ y = \frac{10}{5} = 2 \]
Step 6: Write the final solution
The solution to the system of equations is:
\[ \boxed{(5, 2)} \]
Step 7: Verification
Let's verify our solution by substituting \( x = 5 \) and \( y = 2 \) back into both original equations:
Equation 1: \[ 2(5) + 5(2) = 10 + 10 = 20 \quad \text{(True)} \]
Equation 2: \[ 3(5) - 5(2) = 15 - 10 = 5 \quad \text{(True)} \]
Since both equations are satisfied, the solution \( (5, 2) \) is correct.