Question

Solve the system of equations using the linear combination method.

{5m+3n=413m−6n=9

Answer 1

To solve the system of equations using the linear combination method, we have the following equations:

1. \( 5m + 3n = 41 \) (Equation 1)
2. \( 3m - 6n = 9 \) (Equation 2)

### Step 1: Align the equations for elimination

To eliminate one variable, we can manipulate the coefficients. Let's eliminate \(n\).

We can multiply Equation 1 by 2 to align the coefficients of \(n\):

\[
2(5m + 3n) = 2(41)
\]

This gives us:

\[
10m + 6n = 82 \quad \text{(Equation 3)}
\]

Now we will keep Equation 2 as is:

\[
3m - 6n = 9 \quad \text{(Equation 2)}
\]

### Step 2: Add Equation 3 and Equation 2

Next, we can add Equation 3 and Equation 2 to eliminate \(n\):

\[
10m + 6n + 3m - 6n = 82 + 9
\]

This simplifies to:

\[
(10m + 3m) + (6n - 6n) = 82 + 9
\]

\[
13m = 91
\]

### Step 3: Solve for \(m\)

To find \(m\), divide both sides by 13:

\[
m = \frac{91}{13} = 7
\]

### Step 4: Substitute \(m\) back into one of the original equations

Now that we have \(m\), substitute \(m = 7\) back into Equation 1 to find \(n\):

\[
5(7) + 3n = 41
\]

This gives:

\[
35 + 3n = 41
\]

Subtract 35 from both sides:

\[
3n = 6
\]

Now divide by 3:

\[
n = 2
\]

### Step 5: Write the solution

The solution to the system of equations is:

\[
m = 7, \quad n = 2
\]

Thus, the final answer is:

\[
(m, n) = (7, 2)
\]