A linear model connects a starting amount to a constant rate of change. The challenge is often choosing the right equation before doing any arithmetic. Define the unknown with its units, identify fixed and changing amounts, and decide whether the situation describes an equality or a limit.
A reliable method
Name the unknown
Write what x represents, such as visitors, minutes or the number of tickets. A price and a number of items cannot be added unless the items have first been converted to a cost.
Translate the relationship
A fixed fee plus a rate becomes b + mx. ‘At most’ becomes ≤ and ‘at least’ becomes ≥. If the entire bill is discounted, apply the multiplier to the entire bracket.
Keep both sides balanced
Undo addition before division. For a system, elimination subtracts or adds whole equations; substitution replaces an unknown with an equivalent expression.
Check the meaning
Substitute the answer into the original equation. A maximum number of people must be a whole number that stays within the budget. The slope has units of output per unit of input.
Worked examples
A club pays $18 plus $7 per visitor. A 10% discount on the whole bill makes the payment $79.20. How many visitors attended?
- Let v be the number of visitors. The discounted bill is 0.9(18 + 7v) = 79.2.
- Undo the discount: 18 + 7v = 79.2 ÷ 0.9 = 88.
- Subtract 18 and divide by 7: v = 70 ÷ 7 = 10.
- Check: 0.9(18 + 70) = 79.2.
Answer: 10 visitors
Adult tickets cost $12 and student tickets cost $8. A group buys 25 tickets for $248. How many adult tickets are there?
- Let a and s be the ticket counts. Then a + s = 25 and 12a + 8s = 248.
- Multiply the first equation by 8: 8a + 8s = 200.
- Subtract from the cost equation: 4a = 48, giving a = 12.
- The remaining 13 tickets cost $104; $144 + $104 = $248.
Answer: 12 adult tickets
Now use the method yourself.
Mark this guide as read, then use the method in topic practice before the checkpoint.