Question

3x + 5y = 5
-2x - 8y = 6

Answer 1

To solve the system of equations:

1. \( 3x + 5y = 5 \) (Equation 1)
2. \( -2x - 8y = 6 \) (Equation 2)

We can use either substitution or elimination. Here, I will use the elimination method.

**Step 1: Multiply Equation 1 to match the coefficients of \(y\) in both equations.**

We can multiply Equation 1 by 2 to eliminate \(y\):

\[
2(3x + 5y) = 2(5)
\]
\[
6x + 10y = 10 \quad \text{(Equation 3)}
\]

**Step 2: Modify Equation 2.**

Equation 2 can be left as is:

\[
-2x - 8y = 6 \quad \text{(Equation 2)}
\]

**Step 3: Add Equation 2 and Equation 3 to eliminate \(y\).**

Now we add Equation 3 and Equation 2:

\[
(6x + 10y) + (-2x - 8y) = 10 + 6
\]

This simplifies to:

\[
(6x - 2x) + (10y - 8y) = 16
\]
\[
4x + 2y = 16
\]

**Step 4: Solve for \(x\).**
From \(4x + 2y = 16\):

Divide the entire equation by 2:

\[
2x + y = 8 \quad \text{(Equation 4)}
\]

**Step 5: Substitute \(y\) from Equation 4 back into Equation 1.**

From Equation 4, express \(y\):

\[
y = 8 - 2x
\]

Now substitute \(y\) into Equation 1:

\[
3x + 5(8 - 2x) = 5
\]
\[
3x + 40 - 10x = 5
\]
\[
-7x + 40 = 5
\]

**Step 6: Solve for \(x\).**

Subtract 40 from both sides:

\[
-7x = 5 - 40
\]
\[
-7x = -35
\]

Now divide by -7:

\[
x = 5
\]

**Step 7: Solve for \(y\) using the value of \(x\).**

Substitute \(x = 5\) back into Equation 4:

\[
y = 8 - 2(5)
\]
\[
y = 8 - 10
\]
\[
y = -2
\]

**Final Solution:**

The solution to the system of equations is:

\[
(x, y) = (5, -2)
\]

To verify, substitute \(x = 5\) and \(y = -2\) back into the original equations to ensure they hold true.

1. \(3(5) + 5(-2) = 15 - 10 = 5\) ✔️
2. \(-2(5) - 8(-2) = -10 + 16 = 6\) ✔️

Both equations are satisfied. Thus, the solution is correct.