# The linear model in money matters · Grade 10 · 3 units

> Explanation, worked examples and free practice following the Israeli curriculum.

In The linear model in money matters (Grade 10 · 3 units) you practise 4 skills: The rate and the starting value · Matching a story, a rule and a graph · Two plans: parallel, crossing or the same · Where two graphs cross.

Many charges are made of two parts: a fixed amount paid once, and an amount for each unit, such as a minute, a kilometre or a month. A rule like that is a straight line.

The number multiplying the variable is the rate: how much the price goes up for each unit. The number added on its own is the starting value: the price before any unit is used. On a graph, the starting value is where the line meets the vertical axis.

Two plans with the same rate and different starting values are parallel lines: they never cost the same. Two plans with different rates are lines that cross: they cost the same once.

To find when the price is the same, set the two rules equal and solve the equation. After that point, the plan with the smaller rate is cheaper.

**Tip:** Before you choose a plan, check how many units you will really need: the cheaper plan depends on it.

## What you learn

### The rate and the starting value

- Example: A monthly phone bill is made up of a fixed charge and a charge for each minute of calls. The rule is y = 0.5x + 50 , where x is the number of minutes and y is the bill in shekels. What is the fixed charge?
- Answer: 50 shekels
- Solution: The number added on its own is 50, so the answer is 50 shekels.

### Matching a story, a rule and a graph

- Example: A monthly phone bill is made up of a fixed charge and a charge for each minute of calls. The fixed charge is 50 shekels, and each minute of calls costs 0.50 shekels. Which rule gives y , the bill, from x , the number of minutes?
- Options: y = 50x + 0.5 · y = x + 50 · y = 0.5x · y = 0.5x + 50
- Answer: y = 0.5x + 50
- Solution: The rate is 0.5 and the starting value is 50, so y = 0.5x + 50 .

### Two plans: parallel, crossing or the same

- Example: There are two plans for hiring a bike. In both plans x is the number of hours and y is the price in shekels. Plan A: y = 3x + 35 . Plan B: y = 35 + 3x . What is true of the graphs of the two plans?
- Options: They are parallel: the price is never the same · They cross: the price is the same once · They are the same line: it is the same plan
- Answer: They are the same line: it is the same plan
- Solution: Both have the rate 3 and the start 35. It is the same plan, written in a different order.

### Where two graphs cross

- Example: The graphs show the price at two gyms, A and B , against the number of months. (Two straight lines of price against months, labelled with letters, that cross at one point) At which point is the price at the two gyms the same?
- Answer: (2, 100)
- Solution: The lines meet at the point (2, 100) .

## Not on the site yet

- Explaining a rate of change in words
- Fitting a model to real data

The exercises are made by rules and the answers are worked out in code. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 3 units (grades 10 and 11).

[Practise on the site](https://kixplo.com/il/en/) · [All Grade 10 · 3 units topics](https://kixplo.com/il/en/math/grade-10-3u/)