2. Budgeting and Saving

Budget questions are linear equations in disguise. You usually know a starting amount and a steady weekly or monthly change.

Worked example (saving up): You have $80 saved and add $25 every week. How many weeks until you reach $330 for a phone?

Let \(w\) be the number of weeks. Savings grow from 80 by 25 each week:
\[ 80 + 25w = 330 \]
Subtract 80: \(25w = 250\). Divide by 25: \(w = 10\) weeks.

Worked example (spending down): You start the month with $600 and spend about $45 per day. The amount left after \(d\) days is
\[ A = 600 - 45d. \]
After 9 days: \(A = 600 - 45(9) = 600 - 405 = 195\) dollars left.

Notice the pattern: a starting value plus a constant rate of change is \(y = \text{start} + (\text{rate})x\). A positive rate grows; a negative rate shrinks. This is the same as the slope-intercept line \(y = mx + b\).