Question

Solve the system of equations below using elimination. You must show all work to earn full credit.



2x+5y=20


3x−5y=5

Answer 1

To solve the system of equations using the elimination method, we'll start with the given equations:

1) \( 2x + 5y = 20 \) \quad (Equation 1)

2) \( 3x - 5y = 5 \) \quad (Equation 2)

### Step 1: Align the equations
We will align the equations for clarity.

\[
\begin{align*}
1) & \quad 2x + 5y = 20 \\
2) & \quad 3x - 5y = 5 \\
\end{align*}
\]

### 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.