James and Lin got married and got new jobs. Lin earns $3400 more per year than James. Together they earn $82,300. How much does each of them earn per year?

Let James' yearly earnings be represented by $ J $, and Lin's yearly earnings by $ L $.

We know the following:
1. Lin earns $3400 more than James:  
   $ L = J + 3400 $
2. Together they earn $82,300:  
   $ J + L = 82,300 $

### Step-by-step solution:

Substitute the first equation $ L = J + 3400 $ into the second equation $ J + L = 82,300 $:

\[
J + (J + 3400) = 82,300
\]

Simplify:

\[
2J + 3400 = 82,300
\]

Subtract 3400 from both sides:

\[
2J = 78,900
\]

Divide by 2:

\[
J = 39,450
\]

Now, substitute $ J = 39,450 $ into the first equation to find Lin's earnings:

\[
L = 39,450 + 3400 = 42,850
\]

### Final Answer:
- James earns $39,450 per year.
- Lin earns $42,850 per year.

Would you like further details or have any questions?

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Here are 5 related questions to explore:

1. If Lin earned twice as much as James, what would their combined earnings be?
2. What percentage of their total income does James earn?
3. If both receive a 5% raise, how much will each earn per year after the raise?
4. If their total earnings next year increase by $10,000, how should this amount be divided between them based on their current earnings ratio?
5. If Lin's salary was decreased by $5000, how would that affect their combined income?

**Tip:** When solving problems with relationships between quantities, substituting one equation into another can simplify the process significantly!

Discover how to solve a linear equation involving two variables with this step-by-step guide. James earns $39,450, and Lin earns $42,850 per year, together making $82,300. Learn how to set up the problem, apply algebraic methods, and find the solution. Suitable for Grades 7-9.

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