Compare linear relations and interpret their intersection

Ontario Grade 9 Mathematics

Algebra

Ontario MTH1W · Expectation C3.3

You have probably had to choose between two options that both depend on the same thing — time, distance, or number of items. Maybe you wondered: at what point does one phone plan become cheaper than another? Or when does one job start paying more than a second job? These are exactly the kinds of questions that comparing two linear relations can answer. When two straight-line relationships share a common input value that gives the same output value, they intersect. Finding and explaining that intersection point is the heart of this lesson.

What you will learn

Explain what it means for two linear relations to intersect.
Find the point of intersection by reading a table of values and by using an equation.
Interpret the point of intersection in a real-world context.
Compare two linear relations to decide which gives a better outcome for a given situation.

Prerequisite Review: What Is a Linear Relation?

A linear relation is a relationship between two quantities where the rate of change is constant. On a graph, it forms a straight line. You can write it in the form

y = mx + b

, where

m

is the rate of change (how much

y

changes for every increase of 1 in

x

) and

b

is the initial value (the value of

y

when

x = 0

).

For example, if a streaming service charges a CAD 5 monthly fee plus CAD 2 per movie rented, the total cost

C

after renting

n

movies is

C = 2n + 5

. The rate of change is 2 (cost goes up by CAD 2 per movie) and the initial value is 5 (you pay CAD 5 even before renting anything).

You will need to substitute values into an equation and solve a one-step or two-step equation — both skills from Grade 8. We will review them inside each example as they come up.

y = mx + b

A linear relation has a constant rate of change.
The equation $y = mx + b$ describes a linear relation; $m$ is the rate of change and $b$ is the initial value.
Substituting a known value of $x$ lets you calculate the matching value of $y$ .

Comparing Two Linear Relations Using a Table

When you have two linear relations that share the same input variable, you can build a table of values for both and look for the row where their output values are equal. That shared row is the intersection.

Consider two tutoring services. Service A charges CAD 10 per hour with no registration fee:

C_A = 10h

. Service B charges CAD 6 per hour but has a CAD 20 registration fee:

C_B = 6h + 20

. Both costs depend on the same variable — the number of hours

h

.

By calculating both costs for several values of

h

, you can see exactly when they are equal and what happens before and after that point. Before the intersection, one service is cheaper; after it, the other is. The table in this lesson shows this clearly. Once you spot the equal row, you have found the intersection point.

C_A = 10h, \quad C_B = 6h + 20

Build a table with the same input values for both relations.
Look for the row where both output values are identical — that is the intersection.
Rows before and after the intersection show which option is cheaper or better.

The Point of Intersection: Meaning and Notation

The point of intersection is the one input value at which both linear relations produce the same output value. On a graph, it is the single point where the two lines cross each other.

We write a point as an ordered pair

(x, y)

, where

x

is the input and

y

is the output. The first number tells you the input (for example, the number of hours), and the second number tells you the shared output (for example, the cost in dollars at that moment).

At the intersection, the two relations are equal, meaning

m_1 x + b_1 = m_2 x + b_2

holds true for that specific value of

x

. Before that point, one relation gives a smaller output. After that point, the other relation gives a smaller output. This is what comparing the two relations really means — understanding which is better and when.

m_1 x + b_1 = m_2 x + b_2

The point of intersection is written as the ordered pair $(x, y)$ .
At the intersection, both relations give the exact same output for the same input.
The intersection separates the regions where one relation is greater or less than the other.

Finding the Intersection Symbolically

A table works well for whole-number answers, but sometimes the intersection falls between whole numbers, or the numbers get large. In those cases, you can find the intersection by setting the two expressions equal to each other and solving the equation.

The logic is simple: the intersection is the value of

x

where both relations give the same

y

. So you write

m_1 x + b_1 = m_2 x + b_2

and solve for

x

using inverse operations — the same strategy you used in Grade 8 to solve two-step equations.

Once you find

x

, substitute it back into either original equation to find the matching

y

-value. Write the answer as an ordered pair and always interpret it in the context of the problem. A number without a meaning is incomplete.

m_1 x + b_1 = m_2 x + b_2

Set the two expressions equal: $m_1 x + b_1 = m_2 x + b_2$ .
Collect variable terms on one side and constant terms on the other using inverse operations.
Substitute the found $x$ -value back into either equation to get $y$ .
State the answer as an ordered pair and explain what it means in context.

Interpreting the Intersection and Comparing Regions

Finding the intersection point is only the first part of the job. The second — and equally important — part is interpreting it. Ask yourself: What does the

x

-value represent? What does the

y

-value represent? What decision should someone make based on this information?

For example, if the intersection of two cost relations is at

(5, 50)

, it means that at 5 hours (or 5 items, or whatever the input measures), both options cost exactly CAD 50. For fewer than 5 hours, you compare the two equations by testing a value less than 5 to see which gives a smaller cost. For more than 5 hours, test a value greater than 5.

Comparing does not stop at the intersection. Always describe what happens on each side of it. That full comparison — which is better before, which is equal at, and which is better after — is what expectation C3.3 is asking you to communicate.

y_1 < y_2 when x < x_{intersection}, y_1 > y_2 when x > x_{intersection}

Interpret both coordinates of the intersection point using the units of the problem.
Test a value less than the intersection $x$ to compare the relations before it.
Test a value greater than the intersection $x$ to compare the relations after it.
State a clear recommendation based on the comparison.

Cost Comparison Table: PrintFast vs. BannerPro

Number of Banners ( $n$ )	PrintFast Cost $C_P = 8n$ (CAD)	BannerPro Cost $C_B = 5n + 18$ (CAD)	Cheaper Option
1	8	23	PrintFast
3	24	33	PrintFast
6	48	48	Equal (intersection)
8	64	58	BannerPro
10	80	68	BannerPro

Worked example

Choosing Between Two Delivery Services

A school club needs to order printed banners. PrintFast charges CAD 8 per banner with no setup fee. BannerPro charges CAD 5 per banner plus a CAD 18 setup fee. (a) Write a linear equation for each service. (b) Find the point of intersection symbolically. (c) Interpret the intersection and state which service the club should choose if they need 8 banners.

Step 1 — Write the equations
Let $n$ be the number of banners and $C$ be the total cost in CAD. PrintFast has a rate of change of 8 and no initial fee, so its equation is found by using $C = mn + b$ with $m = 8$ and $b = 0$ . BannerPro has a rate of change of 5 and an initial value of 18.
$C_P = 8n, \quad C_B = 5n + 18$
Step 2 — Set the expressions equal
At the intersection, both services charge the same total cost. So we set $C_P$ equal to $C_B$ and write one equation with one unknown.
$8n = 5n + 18$
Step 3 — Collect variable terms on one side
Subtract $5n$ from both sides to move all $n$ terms to the left. This uses the inverse operation of addition.
$8n - 5n = 18$
Step 4 — Simplify the left side
Combine like terms on the left side.
$3n = 18$
Step 5 — Solve for $n$
Divide both sides by 3 to isolate $n$ .
$n = 6$
Step 6 — Find the matching cost
Substitute $n = 6$ into the PrintFast equation. You can use either equation — the result must be the same.
$C_P = 8(6) = 48$
Step 7 — Verify using BannerPro
Substitute $n = 6$ into the BannerPro equation to confirm the costs match.
$C_B = 5(6) + 18 = 30 + 18 = 48 \checkmark$
Step 8 — Write and interpret the intersection
The intersection is the ordered pair $(6, 48)$ . This means that when the club orders exactly 6 banners, both services cost exactly CAD 48. For fewer than 6 banners, test $n = 2$ : $C_P = 16$ and $C_B = 28$ , so PrintFast is cheaper. For more than 6 banners, test $n = 8$ : $C_P = 64$ and $C_B = 58$ , so BannerPro is cheaper.
$(n, C) = (6, 48)$

Answer: The point of intersection is

(6, 48)

. At 6 banners, both services cost CAD 48. Since the club needs 8 banners, which is more than 6, BannerPro is the better choice at CAD 58 compared to CAD 64 from PrintFast.

Check: Substituting

n = 6

into both equations gives

C_P = 48

and

C_B = 48

. The costs match, confirming the intersection is correct. Checking

n = 8

confirms BannerPro saves the club CAD 6.

Common mistakes and how to avoid them

Writing the intersection as just a single number, such as

n = 6

, instead of as an ordered pair

(6, 48)

.

Correction: The intersection is a point with two coordinates. Always find both the input value and the matching output value, then write them as

(x, y)

.

Forgetting to interpret the intersection in context — stating only the numbers without explaining what they mean.

Correction: Always connect both coordinates to the units of the problem. For example: 'At 6 banners, both services cost the same amount, CAD 48.'

Assuming the cheaper-before option is always cheaper after the intersection too.

Correction: The intersection is where the cheaper option switches. Always test a value on each side of the intersection to see which relation is smaller in each region.

When setting the two expressions equal, subtracting the variable term incorrectly and ending up with a negative coefficient.

Correction: Subtract the smaller variable term from both sides so the coefficient stays positive. For

8n = 5n + 18

, subtract

5n

(not

8n

) to get

3n = 18

, which is easier to work with.

Finding

x

at the intersection but forgetting to substitute back to find

y

.

Correction: After finding

x

, always substitute it back into one of the original equations to calculate

y

. The intersection has two coordinates and you need both.

Lesson summary

A linear relation has a constant rate of change and can be written as $y = mx + b$ , where $m$ is the rate of change and $b$ is the initial value.
Two linear relations can be compared by building a table of values for the same input values and looking for where their outputs are equal.
The point of intersection is the ordered pair $(x, y)$ where both linear relations produce exactly the same output for the same input.
To find the intersection symbolically, set the two expressions equal — $m_1 x + b_1 = m_2 x + b_2$ — and solve for $x$ using inverse operations, then substitute back to find $y$ .
Before the intersection, one relation gives a smaller output; after it, the other does. Always test values on both sides to complete the comparison.
Interpret the intersection using the units and context of the problem, and use it to make a clear, reasoned decision.

Check your understanding

Question 1

Two catering companies charge the following amounts for meals: Company X charges

C_X = 12n + 20

and Company Y charges

C_Y = 16n

, where

n

is the number of meals. At what number of meals do the two companies charge the same total cost?

$n = 4$
$n = 5$
$n = 6$
$n = 8$

Show answer and explanation

n = 5

Set the expressions equal:

12n + 20 = 16n

. Subtract

12n

from both sides to get

20 = 4n

. Divide both sides by 4 to get

n = 5

. Checking:

C_X = 12(5) + 20 = 80

and

C_Y = 16(5) = 80

. Both equal CAD 80, confirming

n = 5

is correct.

Question 2

Using the same catering companies (

C_X = 12n + 20

and

C_Y = 16n

), which company is cheaper for 3 meals?

Company X, because its rate of change is smaller.
Company Y, because it has no initial fee.
Company X, because $C_X = 56$ and $C_Y = 48$ at $n = 3$ .
Company Y, because $C_Y = 48$ and $C_X = 56$ at $n = 3$ .

Show answer and explanation

Company Y, because

C_Y = 48

and

C_X = 56

at

n = 3

.

Substitute

n = 3

:

C_X = 12(3) + 20 = 36 + 20 = 56

and

C_Y = 16(3) = 48

. Since

48 < 56

, Company Y is cheaper at 3 meals. The intersection is at

n = 5

, and 3 is less than 5, so Company Y (with no setup fee) is still the lower-cost option in this region.

Question 3

The intersection of two linear relations is the point

(4, 30)

. What does this tell you?

The two relations are equal for all values of $x$ .
When the input is 4, both relations give an output of 30.
When the output is 4, both relations give an input of 30.
One relation always gives a higher output than the other.

Show answer and explanation

When the input is 4, both relations give an output of 30.

An ordered pair

(x, y)

means the input is

x

and the output is

y

. The point

(4, 30)

means that when

x = 4

, both relations produce

y = 30

. They are equal only at that one point, not everywhere.

Question 4

Relation 1 is

y = 3x + 6

and Relation 2 is

y = 7x - 2

. What is the

y

-value at their point of intersection?

$y = 2$
$y = 12$
$y = 14$
$y = 16$

Show answer and explanation

y = 12

Set the expressions equal:

3x + 6 = 7x - 2

. Subtract

3x

from both sides:

6 = 4x - 2

. Add 2 to both sides:

8 = 4x

. Divide by 4:

x = 2

. Substitute into Relation 1:

y = 3(2) + 6 = 6 + 6 = 12

. Verify with Relation 2:

y = 7(2) - 2 = 14 - 2 = 12

. The

y

-value at the intersection is 12.

Key terms

Linear relation: A relationship between two quantities where the rate of change is constant, forming a straight line when graphed. It can be written as $y = mx + b$ .
Rate of change: How much the output ( $y$ ) increases or decreases for every increase of 1 in the input ( $x$ ). It is represented by $m$ in $y = mx + b$ .
Initial value: The output value when the input is zero. It is represented by $b$ in $y = mx + b$ and is where the line crosses the vertical axis.
Point of intersection: The single point where two linear relations produce the same output for the same input. It is written as an ordered pair $(x, y)$ .
Ordered pair: A pair of numbers written as $(x, y)$ that identifies a specific point. The first number is the input and the second number is the output.
Inverse operation: The opposite operation used to undo a calculation and isolate a variable. For example, division undoes multiplication; subtraction undoes addition.
Substitution: Replacing a variable in an equation with a known number to calculate the value of another unknown quantity.
Compare (linear relations): To examine two linear relations together to determine when one gives a greater, lesser, or equal output compared to the other, including identifying and interpreting their intersection.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 9 Mathematics (MTH1W), expectation C3.3. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

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C3.3 · Compare linear relations and interpret their intersection

Ontario Grade 9 Mathematics

Ontario MTH1W · Expectation C3.3

What you will learn

Prerequisite Review: What Is a Linear Relation?

Comparing Two Linear Relations Using a Table

The Point of Intersection: Meaning and Notation

Finding the Intersection Symbolically

Interpreting the Intersection and Comparing Regions

Cost Comparison Table: PrintFast vs. BannerPro

Choosing Between Two Delivery Services

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MTH1W

About this lesson and its review