To solve this system of equations, we follow a step-by-step algebraic approach:
### Step 1: Write down the given equations.
We have two equations:
1. [tex]\( \frac{m}{n} = \frac{6}{11} \)[/tex]
2. [tex]\( n - m = 155 \)[/tex]
### Step 2: Express [tex]\(m\)[/tex] in terms of [tex]\(n\)[/tex] from the first equation.
From the first equation:
[tex]\[ \frac{m}{n} = \frac{6}{11} \][/tex]
We can cross-multiply to eliminate the fraction:
[tex]\[ 11m = 6n \][/tex]
This rearranges to:
[tex]\[ m = \frac{6}{11}n \][/tex]
### Step 3: Substitute [tex]\(m\)[/tex] from Step 2 into the second equation.
The second equation is:
[tex]\[ n - m = 155 \][/tex]
Substitute [tex]\( m = \frac{6}{11}n \)[/tex] into this equation:
[tex]\[ n - \frac{6}{11}n = 155 \][/tex]
### Step 4: Simplify the equation to find [tex]\(n\)[/tex].
Combine like terms:
[tex]\[ \frac{11n}{11} - \frac{6n}{11} = 155 \][/tex]
[tex]\[ \frac{5n}{11} = 155 \][/tex]
Solve for [tex]\(n\)[/tex]:
[tex]\[ 5n = 155 \times 11 \][/tex]
[tex]\[ 5n = 1705 \][/tex]
[tex]\[ n = \frac{1705}{5} \][/tex]
[tex]\[ n = 341 \][/tex]
### Step 5: Substitute [tex]\(n\)[/tex] back into the equation for [tex]\(m\)[/tex].
We found that [tex]\( n = 341 \)[/tex]. Now, use the expression [tex]\( m = \frac{6}{11}n \)[/tex]:
[tex]\[ m = \frac{6}{11} \times 341 \][/tex]
[tex]\[ m = 186 \][/tex]
### Conclusion
The values of [tex]\(m\)[/tex] and [tex]\(n\)[/tex] that satisfy both equations are:
[tex]\[ m = 186 \][/tex]
[tex]\[ n = 341 \][/tex]
