Direct variation is a straight line through the origin. Double one quantity and the other doubles, and the constant of proportionality is just the gradient with a job to do.
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
If 4 identical textbooks weigh 6 kg, how much do 10 weigh? Work it out any way you like, then say what stayed constant while you did it.
Before you work it out, what is your instinct? Write it down, then check it against the lesson.
Work through the core explanation before applying it.
"$y$ varies directly with $x$" means $y = kx$ for some non-zero constant $k$. The graph is a straight line through the origin, because zero of one quantity gives zero of the other. One known pair of values is enough to find $k$.
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
Work through the core explanation before applying it.
A description such as "the weight varies directly with the number of books" translates straight to $w = kb$, where $k$ is the weight of one book.
Substitute the known pair to find $k$. With 4 books weighing 6 kg: $6 = k \times 4$, so $k = 1.5$. The completed model is $w = 1.5b$.
Only now answer the question. For 10 books: $w = 1.5 \times 10 = 15$ kg. Finding $k$ first means every later part of the question is a single substitution.
"Varies directly" means $y = kx$. Substitute the given pair to find $k$, write the completed formula, then answer the question. The constant is a rate and carries units.
Pause, copy the translation from words to $y = kx$, the worked books example giving $k = 1.5$ kg per book, and the find-$k$-first order, into your book.
We just saw how to build the model from one pair. That raises a question: what does direct variation look like on a graph, and how is it different from any other straight line? This card answers it → it is the straight line whose intercept is zero.
If $y = kx$ then substituting $x = 0$ gives $y = 0$. So the graph must pass through the origin: zero books weigh zero kilograms, zero hours earn zero pay.
That is what separates direct variation from a general linear model. $y = 3x$ varies directly; $y = 3x + 5$ does not, because at $x = 0$ there is already 5 of $y$ present.
So a cost with a call-out fee is **not** direct variation, even though it is linear. The fixed fee breaks the proportionality.
$y = kx$ passes through the origin, since zero of one quantity gives zero of the other. A linear model with a non-zero constant term is not direct variation, so a cost with a fixed fee is linear but not proportional.
Pause, copy the origin argument, and the contrast between $y = 3x$ and $y = 3x + 5$, into your book.
We just saw what direct variation looks like as a graph. That raises a question: given a table of measurements rather than a description, how do you tell? This card answers it → divide, and see whether you get the same number every time.
If $y = kx$ then $\frac{y}{x} = k$ for every pair. So dividing each $y$ by its $x$ should give the same value throughout.
For books: $\frac{6}{4} = 1.5$ and $\frac{15}{10} = 1.5$. Constant, so the data is consistent with direct variation and $k = 1.5$.
If the ratios differ, the relationship is not direct variation. Data giving 1.5, 1.5 and then 2.1 is either not proportional or contains an error worth questioning.
Test data by dividing: $\frac{y}{x}$ is constant exactly when the relationship is direct variation, and that constant is $k$. A constant difference instead signals a linear relationship that may not pass through the origin.
Pause, copy the constant-ratio test with the books data, and the distinction between constant ratio and constant difference, into your book.
Worked examples · 3 in a row, reveal as you go
Follow the reasoning through complete worked solutions.
The weight of a stack of identical books varies directly with the number of books. Four books weigh 6 kg. Find the weight of 10 books.
The cost of fabric varies directly with its length. 3 m costs $\$19.50$. How much fabric can be bought for $\$52$?
Decide whether this data shows direct variation: $x = 2, y = 5$; $x = 6, y = 15$; $x = 10, y = 25$.
Work through the core explanation before applying it.
Run the quick drill and copy the summary into your book.
At the start you found the weight of 10 books from 4 weighing 6 kg, and named what stayed constant. Write that constant with its units, and explain why the graph of this relationship must pass through the origin.
Answer the drill bank and rate your confidence.
Pick your answer, then rate your confidence, that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Write full responses, then check them against the model answers.
Q1. The distance a car travels at constant speed varies directly with time. It travels 216 km in 2.4 hours. Find the constant of proportionality with its units, and the distance travelled in 3.5 hours. (3 marks)
Q2. The mass of a metal rod varies directly with its length. A 40 cm rod has mass 1.2 kg. Find the length of a rod of mass 2.1 kg. (3 marks)
Q3. Explain why a taxi fare of $\$4$ plus $\$2$ per kilometre is not an example of direct variation, even though it is linear. (2 marks)
Practice 1: $k = 3$. Practice 2: $y = 33$. Practice 3: no, at $x = 0$ it gives $y = -1$, not 0, so it does not pass through the origin. Practice 4: $k = 12$ dollars per ticket, so 8 tickets cost $\$96$.
Q1 (3 marks): $d = kt$, and $216 = k \times 2.4$ [1]. $k = 90$ kilometres per hour [1]. $d = 90 \times 3.5 = 315$ km [1].
Q2 (3 marks): $m = kL$, and $1.2 = k \times 40$, so $k = 0.03$ kg per cm [1]. $2.1 = 0.03L$ [1]. $L = 70$ cm [1].
Q3 (2 marks): Direct variation requires that zero of one quantity gives zero of the other, so the graph passes through the origin [1]. A taxi travelling 0 km still costs $\$4$ because of the flagfall, so the relationship is linear with a non-zero intercept and is not direct variation [1].
Take the module quiz if you are ready, then mark the lesson complete.
Build direct-variation models, find the constant and use it both ways. Beat the boss to bank a tier, gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
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