Introduction to Linear Relationships

Learn constant rate of change, initial value.

Lesson 3 of 32 parts Practice Flashcards

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Constant Rate of Change

Objective: Explain constant rate of change, apply the stated method, and check its conditions.

A linear relationship has constant slope m=Δy/Δx when Δx≠0. Equal input changes produce equal output changes. The rate’s units are output units per input unit; a nonzero intercept means the relationship is not direct proportionality. A linear rule is proportional only when its intercept is zero.

Formulas

$m=(y₂−y₁)/(x₂−x₁)$

Worked examples

Try itA line has points (0,7) and (7,49). Find its slope.Show answer

Step 1: Slope is change in y divided by change in x. Step 2: Compute (49−7)/(7−0). Step 3: The slope is 6.

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Constant Rate of Change

Objective: Explain constant rate of change, apply the stated method, and check its conditions.

Formulas

$m=(y₂−y₁)/(x₂−x₁)$

Worked examples

Try itA line has points (0,7) and (7,49). Find its slope.Show answer

Step 1: Slope is change in y divided by change in x. Step 2: Compute (49−7)/(7−0). Step 3: The slope is 6.

Initial Value

Objective: Explain initial value, apply the stated method, and check its conditions.

In y=mx+b, the y-intercept b is the output at x=0. It may represent a starting amount when zero input makes contextual sense. The intercept is distinct from the slope, which describes changes between inputs. A starting value and a rate are different quantities.

Formulas

$y(0)=b$

Worked examples

Try itFor y=6x+7, find y when x=0.Show answer

Step 1: At x=0, the 6x term is zero. Step 2: Then y=0+7. Step 3: The y-intercept is 7.

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