Lesson 4 of 6Linear and Non-Linear Relations

Unit 5 · Lesson 4 of 6

Comparing Two Linear Relations

Compare two linear relations with a table, a graph and equations, find their point of intersection, and explain what it means in a real decision such as choosing a plan.

7 min read8 quick checks11 practice questionsInteractiveAsk AI about this →

Key things to remember

1The point of intersection of two lines is the point that lies on both: the same input gives the same output in both relations.
2To find it with algebra, set the two expressions equal (for example $5 g + 30 = 2 g + 45$ ), solve, then substitute to find the shared value.
3On one side of the intersection one relation has the smaller output; on the other side, the other relation does.
4Different slopes: exactly one intersection. Same slope, different y-intercepts: parallel, no intersection. Same slope and same y-intercept: the same line.
5Always check that the intersection makes sense in the context (no negative times, whole numbers where only whole numbers are possible).

You are choosing a phone plan. Plan A costs 30 dollars a month plus 5 dollars for each gigabyte (GB) of data. Plan B costs 45 dollars a month plus 2 dollars per GB. Plan A starts cheaper, but its cost climbs faster. So which is the better deal? The honest answer is "it depends on how much data you use", and math lets you say exactly where the answer flips. Each plan is a linear relation: Plan A is $C = 5 g + 30$ and Plan B is $C = 2 g + 45$ , where $g$ is the number of GB and $C$ is the monthly cost in dollars. Comparing them means finding the amount of data where they cost the same, and knowing which plan wins on each side of it.

Comparing with a table

List both costs for the same inputs and watch the gap. At 0 GB, Plan A is 15 dollars cheaper. Each extra GB adds 5 dollars to A but only 2 dollars to B, so A's lead shrinks by 3 dollars per GB. After $15 \div 3 = 5$ GB the lead is gone.

Data g (GB)	0	1	2	3	4	5	6	7	8
Plan A: 5g + 30	30	35	40	45	50	55	60	65	70
Plan B: 2g + 45	45	47	49	51	53	55	57	59	61
Cheaper plan	A	A	A	A	A	same	B	B	B

Monthly cost in dollars for each plan

Fig 1Plan B's cost minus Plan A's cost for 0 to 8 GB: Plan A is cheaper by 15, 12, 9, 6 and 3 dollars, the plans cost the same at 5 GB, then Plan B is cheaper by 3, 6 and 9 dollars.

Quick checkTry it before you read on

In the table, how does the gap between the two plans change from one column to the next?

Quick checkTry it before you read on

Suppose Plan A started 21 dollars cheaper than Plan B, with the same rates of 5 and 2 dollars per GB. At how many GB would the plans cost the same?

Comparing with a graph

Graph both relations on the same axes. The point where the lines cross is the point of intersection. It is the one point that lies on both lines, so at that input both relations give the same output. Here it is $(5, 55)$ : at 5 GB, both plans cost 55 dollars. To the left of it, Plan A's line is lower, so A is cheaper. To the right, Plan B's line is lower, so B is cheaper. A graph is quick and shows the whole story at a glance, but reading an intersection that falls between grid lines is only an estimate.

Fig 2Plan A, C = 5g + 30, and Plan B, C = 2g + 45, graphed together; the lines cross at (5, 55), with A lower to the left and B lower to the right

Quick checkTry it before you read on

Two cost lines are graphed on the same axes. For a given input, which option is cheaper?

Quick checkTry it before you read on

Two lines on a graph seem to cross between grid lines, near

(3.4, 21)

. What should you do to get the point of intersection?

VideoGraphing two linear relations on the same axes and reading the point of intersection, the same method this section uses for the two phone plans.Watch on YouTube

Solving with equations

For an exact answer, use the fact that at the intersection the two costs are equal. Set the two expressions equal to each other and solve the equation for $g$ , then substitute $g$ into either equation to get the cost. Substituting into the other equation as well is a free check: both must give the same value.

5 g + 30 3 g + 30 3 g g C = 2 g + 45 = 45 = 15 = 5 = 5 (5) + 30 = 55 and 2 (5) + 45 = 55 ✓

What the point of intersection means

The first coordinate is the input where the two relations are equal (the break-even point: 5 GB). The second coordinate is the value they share there (55 dollars). On one side of the intersection one relation is smaller, and on the other side the other one is. A full answer to "which plan is better?" names both sides.

Quick checkTry it before you read on

The lines

y = 4 x + 10

and

y = x + 22

meet where

4 x + 10 = x + 22

. What is

x

there?

Quick checkTry it before you read on

For the same lines,

y = 4 x + 10

and

y = x + 22

, what is the shared

y

-value at the point of intersection?

VideoTwo phone plans are compared by writing an equation for each and solving to find where the costs are equal, like the Plan A and Plan B example.Watch on YouTube

Lines that never meet, and lines that are the same

Two lines with different slopes cross at exactly one point. Two lines with the same slope but different $y$ -intercepts are parallel: they never meet, so one relation stays ahead by the same amount forever. For example, if two water tanks drain at 20 L per minute and one starts with 500 L while the other starts with 300 L, the first always holds 200 L more. Setting $500 - 20 t = 300 - 20 t$ leads to $500 = 300$ , which is false, and that is how the algebra tells you there is no solution. If the lines have the same slope and the same $y$ -intercept, they are the same line, and every point is shared. Finally, check that an intersection makes sense in context: a crossing at a negative time or at 2.5 concert tickets may not be a real-world answer.

Fig 3Two tanks draining at 20 L per minute, V = 500 − 20t and V = 300 − 20t: the lines are parallel and stay 200 L apart until Tank 2 is empty at 15 minutes, and setting them equal gives the false statement 500 = 300.

Loading interactive…

Turn on compare to graph two options, see where they meet, and read which one is lower on each side of the intersection.

Method	Good for	Watch out for
Table of values	Seeing the gap shrink or grow step by step	The crossing may fall between two rows
Graph	Seeing which option is lower everywhere at once	Reading a crossing between grid lines is only an estimate
Equations	An exact break-even value, even a decimal	Remember to find the shared value and to interpret both sides

Three ways to compare two linear relations

Quick checkTry it before you read on

Solving

4 x - 3 = 4 x - 3

leads to

- 3 = - 3

, which is always true. What does that mean?

Quick checkTry it before you read on

Two plans' cost lines meet at

t = - 2

months. What should you conclude about the real situation?

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1

Plan A costs

C = 5 g + 30

and Plan B costs

C = 2 g + 45

dollars per month for

g

GB of data. Find where they cost the same, and say which plan is cheaper for 8 GB.

Example 2

Gym X charges 20 dollars per visit. Gym Y charges a 120 dollar membership plus 8 dollars per visit. For how many visits do they cost the same? Which is cheaper for 6 visits?

Example 3

Ana walks from a trailhead at 4 km/h. Ben starts at the same moment but 3 km farther along the trail, walking at 2.5 km/h in the same direction. When and where does Ana catch up?

Example 4

Car rental R charges 40 dollars plus 0.25 dollars per km. Rental S charges 25 dollars plus 0.40 dollars per km. For what distance do they cost the same? Which is cheaper for a 150 km trip?

Where marks get lost

Stopping at the intersection. 'They are equal at 5 GB' does not answer 'which plan should I choose?' Say which plan is cheaper below and above that value.
Setting only the rates or only the starting amounts equal (like $5 = 2$ or $30 = 45$ ) instead of setting the whole expressions equal, $5 g + 30 = 2 g + 45$ .
Solving for $g$ and forgetting to substitute back to find the shared cost, the second coordinate of the intersection.
Assuming the option with the smaller starting amount is always cheaper. Plan A starts lower but loses after 5 GB because its rate is higher.
Thinking that no solution means you made an error. Parallel lines (same rate, different starting amounts) never meet, so the equation has no solution.

Flashcards 7

Point of intersection

Break-even point

Finding an intersection with algebra

Parallel lines

+ 3 more

Practice

Plan P costs

C = 6 g + 18

and Plan Q costs

C = 2 g + 42

dollars for

g

GB. For how many GB do they cost the same?

Using the same plans,

C = 6 g + 18

and

C = 2 g + 42

, what is the monthly cost when both plans cost the same?

dollars

With the same plans,

C = 6 g + 18

and

C = 2 g + 42

, which plan is cheaper for 3 GB?

Two cost lines meet at the point (12, 90), where x is the number of months. What does this point mean?

How many points of intersection do

y = 3 x + 2

and

y = 3 x - 5

have?

Gym X charges 15 dollars per visit. Gym Y charges a 60 dollar membership plus 9 dollars per visit. For how many visits do they cost the same?

visits

What is the name of the point where the graphs of two relations cross?

Put these steps in order for comparing two linear cost plans with equations.

1State which plan is cheaper below and above the break-even value
2Substitute into both equations to find and check the shared cost
3Solve for the input variable
4Write an equation for each plan's cost
5Set the two expressions equal to each other

Match each pair of lines to its number of points of intersection.

Different slopes

Same slope, different y-intercepts

Same slope and same y-intercept

Cyclist A rides at 18 km/h from a starting line. Cyclist B starts at the same time 12 km ahead and rides at 12 km/h in the same direction. After how many hours does A catch B?

Rental R costs 40 dollars plus 0.25 dollars per km, and rental S costs 25 dollars plus 0.40 dollars per km. Which statement is correct?

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