Compare two linear relationships

Comparing two linear relationships involves matching their inputs and outputs and interpreting each relationship’s rate of change (slope) and initial value (y-intercept), whether represented by a table, graph, equation, or verbal description. A learner determines which relationship produces the greater output for a given input, recognizing that a larger slope or initial value alone does not always decide the comparison when the relationships intersect. The focus is on accessible linear rules and comparisons, not abstract parameterized families or formal systems methods.

Detailed Explanation: Compare two linear relationships

To compare two linear relationships:

Use the same input for both relationships.
Find each relationship’s output.
Compare the outputs.
Look at the rate of change (slope) and initial value ( $y$ -intercept) to understand why one may be greater.

Example

Relationship A is given by the equation

y=4x+5

Relationship B is shown in the table:

$x$	$y$
$0$	$12$
$1$	$15$
$2$	$18$
$3$	$21$

The output for Relationship B increases by $3$ each time $x$ increases by $1$ , so its rule is

y=3x+12

The question is: Which relationship gives the greater output when $x=5$ ?

Step 1: Find Relationship A’s output

Substitute $5$ for $x$ :

y=4(5)+5=20+5=25

Relationship A gives an output of $25$ .

Step 2: Find Relationship B’s output

Use Relationship B’s rule:

y=3(5)+12=15+12=27

Relationship B gives an output of $27$ .

Step 3: Compare the outputs

27>25

So, when $x=5$ , Relationship B produces the greater output.

Relationship A has the greater slope, $4$ , but Relationship B starts with a greater initial value, $12$ compared with $5$ . A greater slope does not always mean a greater output. The outputs can change which relationship is greater as the input changes.

Learn by doing: Compare two linear relationships

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Types of Solutions - Graph to Equation Pair

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