A linear model is a straight line with a job. Naming what each variable means, and what values it is allowed to take, is half of what the marks are for.
Today's hook, In a modelling question the algebra is usually the easy part. The marks sit in defining the variables, stating the restrictions, and saying what the answer means in the situation.
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Recall, your gut answer first
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
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Recall, your gut answer first
+5 XP warm-up
A plumber charges a $\$70$ call-out fee plus $\$45$ per hour. Write a rule for the total cost. What does each letter stand for, and what values can the hours sensibly take?
Before you work it out, what is your instinct? Write it down, then check it against the lesson.
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Name it, restrict it, then interpret it
Work through the core explanation before applying it.
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Name it, restrict it, then interpret it
+5 XP to read
Every linear model needs three things beyond the equation: which variable is independent (the one you choose), which is dependent (the one that follows), and what restrictions the situation puts on them. A negative number of hours is meaningless, so $h \geq 0$.
$y = mx + b$: $m$ is the rate of change, $b$ the starting value state variables, restrictions, and the answer in context
The gradient is a rate
In a cost model $m$ is dollars per hour, not just a number. Saying what it means in the context is often its own mark.
The intercept is the starting value
A call-out fee, a joining fee, an initial population. It is the value when the independent variable is zero.
Restrictions come from the situation
Hours cannot be negative; a number of items must be a whole number. State them, do not assume the marker infers them.
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What you'll master
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
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What you'll master
Know
Key facts
A linear model has the form $y = mx + b$, where $m$ is a rate of change and $b$ a starting value.
The independent variable is the input you choose; the dependent variable follows from it.
Restrictions are limits the real situation puts on the variables, such as $h \geq 0$.
Two linear models are compared by solving them simultaneously, algebraically or graphically.
Understand
Concepts
Why the gradient of a model carries units and a meaning, not just a value.
Why stating restrictions is part of a correct answer rather than a formality.
Why the intersection of two models is the point where the two options cost or produce the same.
Can do
Skills
Build a linear model from a worded description and define its variables.
State appropriate restrictions on the variables.
Solve a pair of linear models simultaneously and interpret the result in context.
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Key terms
Linear modelA straight-line rule describing a real situation. Like this: $C = 70 + 45h$ gives the cost of a plumbing job lasting $h$ hours.
Independent variableThe quantity you choose or control, plotted on the horizontal axis. Like this: in $C = 70 + 45h$ the hours $h$ are independent.
Dependent variableThe quantity that follows from your choice, plotted vertically. Like this: the cost $C$ depends on how many hours the job takes.
Rate of changeWhat the gradient means in the situation, carrying units. Like this: in $C = 70 + 45h$ the gradient 45 means 45 dollars per hour.
RestrictionA limit the real situation places on a variable. Like this: $h \geq 0$, because a job cannot last a negative number of hours.
Interpretation in contextStating what the mathematical answer means in the situation, in a sentence. Like this: "the two plans cost the same at 8 hours" rather than just $h = 8$.
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Building the model and naming the variables
Work through the core explanation before applying it.
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Building the model and naming the variables
core concept
Start by writing down what each letter stands for, including units. For the plumber: let $C$ be the total cost in dollars and $h$ the number of hours worked.
The fixed part becomes the constant and the per-unit part becomes the gradient: $C = 70 + 45h$. The gradient is not merely 45, it is 45 dollars per hour.
Then state the restriction. Here $h \geq 0$, because a job cannot take negative time. If the plumber charges in whole hours, $h$ would also have to be a whole number, which is a second restriction.
Define before you calculate. A modelling question that opens with "let $C$ be the cost in dollars and $h$ the hours" is already earning marks. An answer that starts straight at $70 + 45h$ has skipped them.
Quick check: in $C = 70 + 45h$, what does the 45 represent?
Define each variable with its units, identify which is independent and which dependent, put the fixed part in the constant and the per-unit rate in the gradient, then state the restrictions the situation imposes. Defining and restricting carry their own marks.
Pause, copy the plumber model with its full definitions and restriction, and the note that the gradient is a rate with units, into your book.
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Comparing two models
core concept
We just saw how to turn a description into a single model. That raises a question: what if there are two options and you want to know which is better? This card answers it → solve them simultaneously; the intersection is where the two options agree.
Two plans, $C_1 = 70 + 45h$ and $C_2 = 120 + 30h$. Setting them equal finds where they cost the same: $70 + 45h = 120 + 30h$, so $15h = 50$ and $h = \frac{10}{3}$ hours.
Graphically that is the point where the two lines cross. Below it one plan is cheaper; above it the other is. The intersection is the decision point.
Interpret it: for jobs shorter than about 3 hours 20 minutes the first plan is cheaper, and for longer jobs the second is. That sentence is what the question is really asking for.
Answer the question that was asked. "Which plan should they choose?" wants a recommendation with a reason, not just $h = \frac{10}{3}$. State the crossover and which side favours which plan.
Fill the blank: setting $70 + 45h = 120 + 30h$ gives $15h = $ .
Two linear models are compared by solving them simultaneously; the intersection is where the options agree. Either side of it, one option is better. Finish with a sentence saying which, and when.
Pause, copy the two-plan comparison, the algebra giving the crossover, and the interpretation sentence, into your book.
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Justifying the conclusion
core concept
We just saw how to find where two models agree. That raises a question: what does a full-mark answer look like once the algebra is done? This card answers it → a conclusion stated in the words of the situation, with the restrictions respected.
A model only applies where its restrictions hold. If the answer comes out at $h = -2$, the correct response is that the situation never occurs, not that the answer is $-2$.
Round in the direction the context demands. If 5.7 hours of labour must be billed in whole hours, the billed time is 6, not 5.
Say what the numbers mean. "The two plans cost the same after 3 hours and 20 minutes; for shorter jobs Plan A is cheaper" answers the question. "$h = 10/3$" does not.
A negative or fractional answer is information. It usually means the situation described cannot happen, or that the model needs rounding to make sense. Say which, rather than reporting the raw number.
Which is NOT part of a complete modelling answer?
Respect the restrictions: an answer outside them means the situation does not occur. Round the way the context demands. Finish with a sentence in the words of the problem, since a bare value does not answer a modelling question.
Pause, copy the three parts of a complete answer (definitions, restrictions, interpretation) and the note that an out-of-range answer is itself information, into your book.
Worked examples · 3 in a row, reveal as you go
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Work examples end to end
Follow the reasoning through complete worked solutions.
PROBLEM 1 · BUILDING A MODEL
A gym charges a $\$60$ joining fee plus $\$25$ per month. Write a model for the total cost, defining your variables and any restrictions.
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Let $C$ be the total cost in dollars and $m$ the number of months of membership
Define with units before calculating.
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$C = 60 + 25m$
Fixed fee is the constant; the monthly rate is the gradient.
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Restriction: $m \geq 0$, and $m$ is a whole number if billing is monthly
A negative or part month is not meaningful here.
PROBLEM 2 · COMPARING TWO PLANS
Plan A costs $\$70$ plus $\$45$ per hour; Plan B costs $\$120$ plus $\$30$ per hour. Find when they cost the same and advise which to choose.
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$70 + 45h = 120 + 30h$
Set the two models equal.
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$15h = 50$, so $h = \dfrac{10}{3} \approx 3.33$ hours
Solve for the crossover.
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For jobs under about $3$ hours $20$ minutes choose Plan A; for longer jobs choose Plan B
The recommendation is the answer, not the number alone.
PROBLEM 3 · INTERPRETING AN OUT-OF-RANGE ANSWER
A model $P = 500 - 20t$ gives the number of fish in a pond after $t$ weeks. Find when the pond is empty, and comment.
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Set $P = 0$: $500 - 20t = 0$, so $t = 25$ weeks
Solve for the required value.
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The restriction is $0 \leq t \leq 25$, since a negative population is meaningless
Beyond 25 weeks the model stops describing reality.
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The pond is empty after $t = 25$ weeks, and the model should not be used beyond that point
State the limitation as part of the answer.
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Quick-fire practice
Work through the core explanation before applying it.
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Quick-fire practice
+10 XP
A taxi charges $\$4$ plus $\$2$ per km. Write a model and define the variables.
For $C = 60 + 25m$, what does 25 mean in context?
State a sensible restriction for a model where $n$ is a number of tickets sold.
Two plans are $C = 20 + 5x$ and $C = 35 + 2x$. Find where they are equal.
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Revisit the plumber
Run the quick drill and copy the summary into your book.
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Revisit the plumber
At the start you wrote a rule for the plumber charging $\$70$ plus $\$45$ per hour. Check your definitions and restriction against the worked examples, and write one sentence saying what the gradient means to the customer.
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Multiple choice
Answer the drill bank and rate your confidence.
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Multiple choice
+5 XP per correct · +25 XP all-correct
Pick your answer, then rate your confidence, that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
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Short answer
Write full responses, then check them against the model answers.
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Short answer
ApplyBand 44 marks
Q1. A printing company charges a $\$150$ setup fee plus $\$2$ per poster. Write a model, define your variables, state a restriction, and find the cost of 400 posters. (4 marks)
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ApplyBand 54 marks
Q2. Company A charges $\$200$ plus $\$15$ per hour; Company B charges $\$80$ plus $\$25$ per hour. Determine when the costs are equal and advise which company to use for a 15-hour job, justifying your answer. (4 marks)
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UnderstandBand 32 marks
Q3. Explain why stating restrictions on the variables is part of a correct modelling answer, using an example. (2 marks)
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📖 Comprehensive answers (click to reveal)
Practice 1: let $C$ be cost in dollars and $d$ the distance in km; $C = 4 + 2d$, with $d \geq 0$. Practice 2: the cost is 25 dollars per month of membership. Practice 3: $n \geq 0$ and $n$ a whole number. Practice 4: $20 + 5x = 35 + 2x$ gives $3x = 15$, so $x = 5$.
Q1 (4 marks): Let $C$ be the total cost in dollars and $n$ the number of posters [1]. $C = 150 + 2n$ [1]. Restriction: $n \geq 0$ and $n$ a whole number [1]. For $n = 400$: $C = 150 + 800 = 950$ dollars [1].
Q2 (4 marks): $200 + 15h = 80 + 25h$ [1]. $120 = 10h$, so $h = 12$ hours [1]. A 15-hour job is longer than the crossover [1]. Beyond 12 hours the company with the smaller hourly rate is cheaper, so Company A should be used: $200 + 225 = 425$ against $80 + 375 = 455$ [1].
Q3 (2 marks): A model only describes the situation over the values the situation allows, so an answer outside them is not meaningful [1]. For example a cost model $C = 60 + 25m$ requires $m \geq 0$, since a negative number of months does not exist and substituting one would produce a cost lower than the joining fee [1].
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Review and finish
Take the module quiz if you are ready, then mark the lesson complete.
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Boss battle · Model Maker
earn bronze · silver · gold
Build linear models, name variables and restrictions, and find crossover points. Beat the boss to bank a tier, gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.