Can a relationship be proportional if the graph is a straight line but does not go through the origin?

No. Both conditions are required. A straight line means the quantity changes at a steady rate, but passing through the origin is what guarantees the ratio y/x stays constant. A line crossing the vertical axis anywhere other than (0,0) has a fixed amount built in, such as a starting fee or a head start, and its ratios shrink or grow as x increases.

Why isn't it enough to check that the numbers go up by the same amount each time?

Because equal differences describe a steady rate, not equal ratios. Consider 30, 60, 90 paired with 1, 2, 3: the ratios are all 30, so it is proportional. Now consider 40, 70, 100 paired with 1, 2, 3: the differences are still 30 each time, but the ratios are 40, 35, and about 33.3. Proportionality is about multiplication, so you must divide y by x in each row.

Can the constant of proportionality be a fraction, a decimal, or negative?

Yes to all three. The only requirements are that k is a single fixed number and that it is not zero. Relationships like y = 0.75x, y = 2/3x, and y = -4x are all proportional. In real-world Grade 7 problems k is usually positive because quantities like cost, distance, and weight are positive, but the math itself allows negatives.

How many pairs of values do I need to check in a table?

Check all of them. Finding one pair whose ratio differs is enough to prove the relationship is not proportional, but showing that two ratios match does not prove it is — a third row could break the pattern. If you check every row given and all the ratios agree, you can conclude the table shows a proportional relationship.

M7MATH-3.3

Recognizing Proportional Relationships

Learn three ways to tell if two quantities are proportional: test equivalent ratios in a table, check for a straight line through the origin, and inspect the equation.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Recognizing Proportional Relationships, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

You already know how to find a unit rate. Now comes the question that shows up all over Grade 7: are these two quantities actually in a proportional relationship, or do they just look related? A recipe that doubles when you double the batch is proportional. A gym that charges a joining fee plus a monthly rate is not — even though the cost still climbs steadily.

This lesson gives you three reliable tests, one for each way information gets presented: a table, a graph, or an equation. All three are really checking the same single idea, so once you see why they agree, you can pick whichever test matches the information in front of you. You will also learn the two traps that catch the most students: relationships that add a constant instead of multiplying, and lines that are perfectly straight but miss the origin.

What Makes a Relationship Proportional

Two quantities

x

and

y

are in a proportional relationship when every pair of values has the same ratio. In symbols,

\frac{y}{x}

gives the same number for every pair (with

x \neq 0

). That repeated number is called the constant of proportionality.

The key word is multiplying. In a proportional relationship you get

y

from

x

by multiplying by one fixed number, always the same number, no matter how big or small

x

is. If 3 pounds of apples cost 6 dollars, then 1 pound costs 2 dollars, 5 pounds cost 10 dollars, and 0 pounds cost 0 dollars. Every cost is 2 times the number of pounds.

Compare that with a relationship that adds a fixed amount. Suppose a taxi charges a 4 dollar pickup fee plus 2 dollars per mile. For 1 mile you pay 6 dollars, so the ratio is

6:1

. For 3 miles you pay 10 dollars, so the ratio is

10:3

, which is about

3.33

. The ratios changed, so the relationship is not proportional — even though the price rises at a steady rate.

That difference explains why zero matters so much. In a proportional relationship, when

x = 0

the value of

y

must also be

0

: multiplying zero by anything gives zero. Any starting fee, head start, or fixed charge breaks proportionality immediately. Steady growth alone is not enough; the growth has to start from nothing.

Testing a Table of Values

When the information comes as a table, compute

\frac{y}{x}

for every row and compare. If all the quotients match, the relationship is proportional and that common quotient is the constant of proportionality. If even one row disagrees, it is not proportional.

Hours worked $x$	Pay $y$ (dollars)	$\frac{y}{x}$
2	30	15
5	75	15
8	120	15

Every ratio is 15, so pay is proportional to hours, with constant of proportionality 15 dollars per hour.

Now a table that fails:

Months $x$	Total saved $y$ (dollars)	$\frac{y}{x}$
1	60	60
2	90	45
4	150	37.5

The quotients shrink, so this is not proportional. Look closely and you can see why: the saver started with 30 dollars and adds 30 each month.

Here is the mistake to avoid. Many students look only at the differences down a column. In the second table,

y

goes up by 30 for each extra month — perfectly steady — and that tempts people to call it proportional. Steady differences are not the test. The test is equal ratios, and you must divide, not subtract.

One more shortcut worth knowing: if a table contains the pair

(0, 0)

and all other ratios agree, that is consistent with proportionality. But if a table shows

x = 0

paired with any nonzero

y

, you can stop right there — it cannot be proportional.

Reading a Graph

Graph the ordered pairs and ask two questions: do the points lie on a straight line, and does that line pass through the origin

(0,0)

? Both must be yes. A proportional relationship graphs as a straight line through the origin, and nothing else does.

Why the origin? Because

x = 0

forces

y = 0

in any proportional relationship. A line that crosses the

y

-axis at 5 tells you there is a fixed amount of 5 built in before

x

ever grows, so the ratios cannot all be equal.

Why straight? Because a constant ratio means every step of the same size in

x

produces the same step in

y

. If the points curve upward or bend, the ratio is changing.

Graph shape	Proportional?	Why
Straight line through $(0,0)$	Yes	Constant ratio $\frac{y}{x}$ at every point
Straight line crossing $y$ -axis above or below the origin	No	A fixed amount is added
Curve through $(0,0)$	No	Ratio changes as $x$ grows
Scattered points, no line	No	No single ratio

A common source of confusion is a graph window that does not show the origin. If the axes start at 10, a line might look like it heads toward the corner of the picture without actually passing through

(0,0)

. Check the scale on both axes before deciding, or pick two points from the line and test their ratios directly.

Also remember that on a proportional graph, the point where

x = 1

shows the constant of proportionality as its

y

-value. That gives you a fast way to read the unit rate straight off the picture.

Inspecting an Equation

An equation describes a proportional relationship exactly when it can be written in the form

y = kx

where

k

is a fixed nonzero number. Nothing added, nothing subtracted, no

x

in a denominator, no

x^2

Equation	Proportional?	Reason
$y = 7x$	Yes	Already in the form $y = kx$ with $k = 7$
$y = \frac{2}{5}x$	Yes	$k$ may be a fraction
$y = 3x + 4$	No	The $+4$ is a fixed amount added
$y = x^2$	No	Ratio $\frac{y}{x} = x$ changes
$\frac{y}{x} = 9$	Yes	Rearranges to $y = 9x$
$y = 12 - x$	No	Ratio is not constant; graph misses the origin
$2y = 10x$	Yes	Divide both sides by 2 to get $y = 5x$

Notice the last row. An equation can be proportional even if it does not start in the form

y = kx

; what matters is whether it can be rewritten that way using equivalent equations. So do the algebra before you judge.

Where students slip: seeing any equation with a plus sign and assuming it must be non-proportional without checking, or ignoring a small added constant like

y = 5x + 0.5

. That half unit is enough to break proportionality. On the other side, some students reject

y = \frac{3}{4}x

because the constant is a fraction, or reject

y = -2x

because it is negative. Both are proportional;

k

just needs to be one fixed nonzero number.

The three tests always agree, because

y = kx

, equal ratios, and a straight line through the origin are three descriptions of the same relationship.

Choosing the Right Test and Explaining Your Answer

In class you rarely get to pick the format — a problem hands you a table, a graph, a description in words, or an equation, and you use the matching test. But you can always convert between them if that helps.

Given a word description, the fastest move is to look for a fixed starting amount. Phrases like joining fee, delivery charge, deposit, head start, or already has signal an added constant, which means not proportional. Phrases like per, each, for every, and at a rate of describe multiplication and usually point toward proportionality — as long as nothing extra is added on.

Given a table, divide

y

x

in every row. Given a graph, check straightness and the origin. Given an equation, try to write it as

y = kx

.

When you write your reasoning, a complete answer names the test you used and shows the evidence. Instead of "yes, it's proportional," write something like: "Proportional, because

\frac{18}{3} = 6

\frac{30}{5} = 6

, and

\frac{48}{8} = 6

, so every ratio equals 6." For a no, point to the specific pair that breaks the pattern: "Not proportional, because

\frac{9}{2} = 4.5

but

\frac{15}{4} = 3.75

."

One counterexample is enough to prove a relationship is not proportional, but showing a single matching ratio never proves that it is. You need every pair you have to agree. That habit of checking all the data, not just the first convenient pair, is what makes your conclusion trustworthy in the next lesson, where you use

k

to write equations and make predictions.

Key terms

Proportional relationship.: A relationship between two quantities in which the ratio $\frac{y}{x}$ is the same for every pair of corresponding values.
Constant of proportionality.: The fixed number $k$ that every $x$ -value is multiplied by to get the matching $y$ -value; it equals the common ratio $\frac{y}{x}$ .
Equivalent ratios.: Two or more ratios that simplify to the same value, such as $6:3$ , $10:5$ , and $14:7$ , all equal to 2.
Origin.: The point $(0,0)$ on a coordinate plane. Every graph of a proportional relationship passes through it.
Unit rate.: The value of a ratio when the second quantity is 1. In a proportional relationship the unit rate is the constant of proportionality.
Counterexample.: A single pair of values whose ratio differs from the others, which is enough to show a relationship is not proportional.
Additive relationship.: A relationship in which a fixed amount is added rather than multiplied, such as $y = 3x + 4$ . Its graph is straight but misses the origin, so it is not proportional.

Worked example

Two skating rinks post their prices. Rink A charges by the hour with this table. Rink B uses the equation

c = 5h + 8

, where

c

is the total cost in dollars and

h

is the number of hours. Decide whether each rink's cost is proportional to the number of hours, and give the constant of proportionality where one exists.

Hours $h$ at Rink A	Cost $c$ (dollars)
2	13
4	26
6	39
10	65

Start with Rink A, using the table test: divide cost by hours in every row.

\frac{13}{2} = 6.5

\frac{26}{4} = 6.5

\frac{39}{6} = 6.5

\frac{65}{10} = 6.5

All four ratios are equal, so Rink A's cost is proportional to the number of hours. The constant of proportionality is

6.5

, meaning 6 dollars and 50 cents per hour, and the equation is

c = 6.5h

. Notice the constant does not have to be a whole number.

Now Rink B, using the equation test. A proportional relationship must be writable as

y = kx

with nothing added. The equation

c = 5h + 8

has the term

+8

, and there is no way to remove it using equivalent equations, so it is not in the form

c = kh

. Rink B is not proportional.

Check that conclusion with the ratio test to be sure. For

h = 1

c = 5(1) + 8 = 13

, so

\frac{13}{1} = 13

. For

h = 4

c = 5(4) + 8 = 28

, so

\frac{28}{4} = 7

. The ratios 13 and 7 disagree, confirming it is not proportional.

In graph language: Rink A is a straight line through the origin, while Rink B is a straight line that crosses the vertical axis at 8 because of the 8 dollar fee charged before any skating time. Both lines are straight, but only one passes through

(0,0)

Practice questions

Which relationship is proportional?

A table where $x = 3$ pairs with $y = 12$ and $x = 5$ pairs with $y = 18$
A straight line that crosses the $y$ -axis at 4
The equation $y = \frac{5}{8}x$
A table where $x = 0$ pairs with $y = 7$

Answer: The equation $y = \frac{5}{8}x$

Check each one. The first table gives

\frac{12}{3} = 4

but

\frac{18}{5} = 3.6

, so the ratios differ. A line crossing the

y

-axis at 4 does not pass through the origin, so a fixed amount has been added. In the last table,

x = 0

pairs with a nonzero

y

, which is impossible when

y = kx

. The equation

y = \frac{5}{8}x

is already in the form

y = kx

with

k = \frac{5}{8}

; a fractional constant of proportionality is completely fine.

A plant is 4 centimeters tall today and grows exactly 2 centimeters each week. Is the plant's height proportional to the number of weeks? Explain using at least two pairs of values.

Answer: No. After 1 week the height is 6 cm, giving the ratio $\frac{6}{1} = 6$ ; after 3 weeks the height is 10 cm, giving $\frac{10}{3} \approx 3.33$ . Since the ratios are not equal, height is not proportional to weeks.

The steady growth of 2 cm per week makes the graph a straight line, which is why this one fools people. But the plant already had 4 cm of height at week 0, so the graph crosses the vertical axis at 4 instead of passing through the origin. Written as an equation it is

h = 2w + 4

, which cannot be rewritten as

h = kw

. Height would be proportional to weeks only if the plant started at 0 cm.

A table shows that 4 notebooks cost 10 dollars and 10 notebooks cost 25 dollars. Assuming the pattern continues, is cost proportional to the number of notebooks? If so, give the constant of proportionality and write the equation.

Answer: Yes. $\frac{10}{4} = 2.5$ and $\frac{25}{10} = 2.5$ , so the constant of proportionality is 2.5 and the equation is $c = 2.5n$ .

Divide cost by number of notebooks for each pair. Both quotients equal 2.5, so the ratios are equivalent and the relationship is proportional. The constant 2.5 means each notebook costs 2 dollars and 50 cents, which is also the unit rate. Once you have

c = 2.5n

you can predict any cost: 7 notebooks would cost

2.5 \times 7 = 17.5

, or 17 dollars and 50 cents.

FAQ

Can a relationship be proportional if the graph is a straight line but does not go through the origin?: No. Both conditions are required. A straight line means the quantity changes at a steady rate, but passing through the origin is what guarantees the ratio $\frac{y}{x}$ stays constant. A line crossing the vertical axis anywhere other than $(0,0)$ has a fixed amount built in, such as a starting fee or a head start, and its ratios shrink or grow as $x$ increases.
Why isn't it enough to check that the numbers go up by the same amount each time?: Because equal differences describe a steady rate, not equal ratios. Consider 30, 60, 90 paired with 1, 2, 3: the ratios are all 30, so it is proportional. Now consider 40, 70, 100 paired with 1, 2, 3: the differences are still 30 each time, but the ratios are 40, 35, and about 33.3. Proportionality is about multiplication, so you must divide $y$ by $x$ in each row.
Can the constant of proportionality be a fraction, a decimal, or negative?: Yes to all three. The only requirements are that $k$ is a single fixed number and that it is not zero. Relationships like $y = 0.75x$ , $y = \frac{2}{3}x$ , and $y = -4x$ are all proportional. In real-world Grade 7 problems $k$ is usually positive because quantities like cost, distance, and weight are positive, but the math itself allows negatives.
How many pairs of values do I need to check in a table?: Check all of them. Finding one pair whose ratio differs is enough to prove the relationship is not proportional, but showing that two ratios match does not prove it is — a third row could break the pattern. If you check every row given and all the ratios agree, you can conclude the table shows a proportional relationship.

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