Find where two linear models have the same value

Ontario Grade 10 Mathematics

Modelling Linear Relations

Compare two changing quantities using a table, a graph, or equations

Suppose two services charge different amounts as you use them. At first, one may cost less. Later, the other may cost less. The point where their costs match is a useful comparison. In this lesson, you will find that point using linear models. A linear model describes a relationship that changes at a steady rate. You will use familiar ideas about inputs, outputs, tables, and equations.

What you will learn

Explain what it means for two linear models to have the same value.
Find the input and shared output by comparing model values.
Check that a proposed answer works in both models.

1. Grade 9 bridge: inputs, outputs, and steady change

A relationship connects an input to an output. For example, the number of hours worked can be an input, and the money earned can be an output. We often use

x

for the input and

y

for the output.

A linear relationship changes at a steady rate. If a value starts at 5 and increases by 3 each time the input increases by 1, its outputs are 5, 8, 11, and 14. The equal increases show the steady rate.

A linear model can be written as an equation such as

y=3x+5

. Here,

3

is the amount the output increases for each increase of 1 in the input. The number

5

is the output when the input is 0. Substituting an input means replacing

x

with that number and calculating the matching output.

y=mx+b

The input is the value you choose; the output is the value the model gives.
A linear model has a steady rate of change.
Two models have the same value when they give the same output for the same input.

2. What does “the same value” mean?

Compare two plans. Plan A costs CAD 12 to start, then CAD 4 for each hour. Plan B costs CAD 4 to start, then CAD 6 for each hour. Let

h

stand for the number of hours and

C

stand for cost in CAD.

Plan A begins with the higher cost, but its cost rises more slowly. Plan B begins lower, but rises more quickly. We want to find the number of hours when the costs match. That matching point has two parts: the input, or number of hours, and the shared output, or cost.

A table lets you compare both models at the same inputs. Look for a row where the outputs are equal. If the match lies between values in the table, you can test more inputs or use an equation. A graph shows the same idea visually: the matching point is where the two lines cross. The crossing point’s horizontal coordinate is the input, and its vertical coordinate is the shared output.

The values must be compared at the same input. Comparing Plan A at 2 hours with Plan B at 3 hours does not show when the plans have the same cost.

C_A=C_B

Use the same input for both models.
The shared value is the output produced by both models.
A table, a graph, or an equation can help locate the match.

3. Guided example: find the matching cost

Use the two service plans from above. Plan A costs CAD 12 plus CAD 4 per hour. Plan B costs CAD 4 plus CAD 6 per hour. We will compare their costs at several whole-number inputs, then use the model equations to find the exact match.

The table shows that the costs match at 4 hours. The equations confirm the result. To find the input, set the two cost expressions equal because both represent the same cost at the matching time. Solve for the input, then substitute it into either model to find the shared cost.

A solution must include both values. Saying only “4” is incomplete unless you say that it means 4 hours. Saying only “CAD 28” is also incomplete because it does not identify when that cost occurs.

4h+12=6h+4

Set the two model outputs equal to find the matching input.
Substitute the input into either model to find the shared output.
State the input and output with their meanings and units.

4. Practice and check your reasoning

When solving a new comparison, first name the input and output. Write or read each model carefully. If the models are given as equations, set their output expressions equal. Use inverse operations to isolate the input: undo addition or subtraction, then undo multiplication or division. Substitute the result into one model. Finally, check that the other model gives the same output.

If a graph is supplied, read the crossing point. If a table is supplied, compare outputs in matching rows. In either case, explain what each coordinate or table entry means in the situation. Do not report a point without connecting it to the quantities being compared.

Independent practice: Model A gives

y=5x+3

and Model B gives

y=2x+15

. Find the input where they have the same value and then find that value. Check by substituting your input into both equations. The answer should show an input and a shared output, not just a calculation.

A good check is to put your result back into both models. If the outputs differ, the input or arithmetic needs another look. Also check whether your answer makes sense for the context. For example, a negative number of hours would not fit a situation where time is measured after a service begins.

y_1=y_2

Use the model that matches the question and keep the input consistent.
Check the result in both models.
Include units and explain what the input and shared value represent.

Comparing the service plans

Hours, h	Plan A cost, CAD	Plan B cost, CAD
0	12	4
2	20	16
4	28	28
6	36	40

Worked example

Two service plans

Plan A costs CAD 12 to start and CAD 4 per hour. Plan B costs CAD 4 to start and CAD 6 per hour. Find the number of hours when their costs match and state the shared cost.

Write the models
Let $h$ be the number of hours and let $C_A$ and $C_B$ be the costs in CAD. The starting charge is the cost at 0 hours, and the hourly charge is added once for each hour.
$C_A=4h+12,\quad C_B=6h+4$
Compare costs in a table
Calculate both costs for the same number of hours. At 4 hours, each plan costs CAD 28, so the table suggests the matching input and output.
$4(4)+12=28,\quad 6(4)+4=28$
Find the matching input
Set the cost expressions equal because the plans have the same cost at the input we are seeking. Subtract $4h$ from both sides, then subtract 4 from both sides. Divide by 2 to isolate $h$ .
$4h+12=6h+4\quad\Rightarrow\quad 8=2h\quad\Rightarrow\quad h=4$
Find and check the shared cost
Substitute 4 hours into either model. Both models give CAD 28, so the answer is consistent.
$C_A=4(4)+12=28,\quad C_B=6(4)+4=28$

Answer: The plans cost the same after 4 hours. The shared cost is CAD 28.

Check: At 4 hours, Plan A costs CAD 28 and Plan B costs CAD 28. The outputs match.

Common mistakes and how to avoid them

Comparing the models at different inputs.

Correction: Use the same input for both models. A match means equal outputs at one shared input.

Reporting only the input or only the output.

Correction: State both values and identify what each one means in the situation.

Finding the input but not checking the shared output.

Correction: Substitute the input into both models. Their outputs should be equal.

Mixing up the starting amount and the rate.

Correction: The starting amount is the output at input 0. The rate is the amount added for each increase of 1 in the input.

Lesson summary

A linear model describes a relationship with a steady rate of change.
Two models have the same value when they give equal outputs at the same input.
Use a table, graph, or equations to find the match.
For equations, set the output expressions equal, find the input, and substitute to find the shared output.
Check that both models give the same result and state the answer with units.

Check your understanding

Question 1

Model A is

y=3x+8

and Model B is

y=5x

. At what input do they have the same value, and what is that value?

Input 4; shared value 20
Input 2; shared value 10
Input 4; shared value 12
Input 8; shared value 40

Show answer and explanation

Input 4; shared value 20

Set the outputs equal:

3x+8=5x

. This gives

x=4

. Substitution gives

3(4)+8=20

and

5(4)=20

.

Question 2

A graph shows two linear models crossing at the point with coordinates

(6, 18)

. What does this point tell you?

The models have the same value of 6 when the input is 18.
The models have the same output, 18, at input 6.
One model has output 6 and the other has output 18.
The models have the same output only at input 18.

Show answer and explanation

The models have the same output, 18, at input 6.

The first coordinate is the input and the second is the output. At input 6, both models give output 18.

Question 3

Two models give outputs 14 and 14 when the input is 3. What is the shared value?

The input is 14 and the shared value is 3.
The shared value is 3.
The shared value is 14 at input 3.
The models do not match because their inputs are equal.

Show answer and explanation

The shared value is 14 at input 3.

The input is 3, and both outputs are 14. Therefore, 14 is the shared value.

Key terms

Input: The value supplied to a model, often represented by $x$ .
Output: The value produced by a model for a chosen input, often represented by $y$ .
Linear model: A model in which the output changes at a steady rate as the input changes.
Shared value: The output that two models give at the same input.

Continue through MFM2P

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 10 Mathematics (MFM2P), study topic G1. 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.

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