Question

solve using elimination
5x + 4y = - 7
- 5x - 2y = 1
(explain with details)

Answer 1

To solve the system of equations using the elimination method, we are given the following equations:

1. \( 5x + 4y = -7 \) (Equation 1)
2. \( -5x - 2y = 1 \) (Equation 2)

### Step 1: Align the equations
We start by writing the equations together for better visualization:

\[
\begin{align*}
5x + 4y & = -7 \quad \text{(1)} \\
-5x - 2y & = 1 \quad \text{(2)}
\end{align*}
\]

### Step 2: Add the two equations
We can eliminate \(x\) by adding Equation 1 and Equation 2. Notice that the coefficients of \(x\) are opposites (\(5\) and \(-5\)).

\[
(5x + 4y) + (-5x - 2y) = -7 + 1
\]

This simplifies to:

\[
0x + (4y - 2y) = -6
\]

This further simplifies to:

\[
2y = -6
\]

### Step 3: Solve for \(y\)
To find \(y\), divide both sides of the equation by \(2\):

\[
y = \frac{-6}{2} = -3
\]

### Step 4: Substitute \(y\) back into one of the original equations
Now that we have \(y = -3\), we can substitute this value back into either Equation 1 or Equation 2. We will use Equation 1:

\[
5x + 4(-3) = -7
\]

This simplifies to:

\[
5x - 12 = -7
\]

### Step 5: Solve for \(x\)
Next, add \(12\) to both sides of the equation:

\[
5x = -7 + 12
\]

This results in:

\[
5x = 5
\]

Now divide both sides by \(5\):

\[
x = \frac{5}{5} = 1
\]

### Step 6: Write the solution
We have found both \(x\) and \(y\):

\[
x = 1, \quad y = -3
\]

### Final Answer
Thus, the solution to the system of equations is:

\[
(x, y) = (1, -3)
\]

You can check your solution by substituting \(x = 1\) and \(y = -3\) back into both original equations to verify that both are satisfied.