F&A - 12 Non-proportional linear relationships: on-proportional linear relationships can be expressed in the form y = mx + b, where b is not 0, m represents the constant rate of change or slope of the line, and b represents the y-intercept. The graph of a non-proportional linear relationship is a straight line that does not pass through the origin.

Example 12.1 – Non-proportional linear relationships (y = mx + b, where b is not 0):

Ajax Taxicab Company charges a flat fee of $1.00 plus $0.30 per mile to ride in a cab. (Assumption: The flat fee is incurred as soon as you enter the cab.)

Rule in words: To determine the cost of an Ajax Taxicab ride, multiply the number of miles traveled by $0.30, and then add $1.00 (the flat fee) to the product.

Rule in Equation: If y represents the total cost of an Ajax Taxicab ride of x miles, then the relationship can be expressed as an equation in the form of y = mx + b, where m represents the cost per mile ($0.30/mile) and b represents the flat fee ($1.00).

Total Cost = Cost per Mile • Number of Miles + Flat Fee
y = 0.30 • x + 1.00, or y = 0.30x + 1.00

Graph and table showing cost of taxi ride

Example 12.2 – Non-proportional linear relationships (with negative slope):

A 10-inch candle burns at a constant rate of 1 inch per hour.

Rule in words: To determine the height of the candle multiply the number of hours that the candle burns by 1 inch per hour, and subtract the product from the candle’s initial height (10 inches).

Rule in Equation: If y represents the height of the candle after x hours of burning, then the relationship can be expressed as an equation in the form y = mx + b, where m represents the rate at which the candle burns (1 inch per hour) and b represents the initial height of the candle (10 inches).

Height of Candle = Rate at which it Burns • Number of Hours Burned + Initial Height
y = -1 • x + 10, or y = 10 -1x

Graph and table of how quickly a candle burns

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