Mathematical understanding involves translating a context into a linear equation by identifying quantities, assigning variables with appropriate units, and interpreting relationships such as constant rate of change, initial value, and equality. The relationship may be expressed as a one-variable equation or in forms such as , with signs, coefficients, and constants reflecting the situation rather than being inserted mechanically. This scope distinguishes proportional and linear relationships from nonlinear ones and excludes systems, piecewise models, and other advanced cases.
A linear equation describes a situation where a quantity changes at a constant rate. To write one:
A taxi charges a starting fee of $3 plus $2 for each mile traveled. Mia’s ride costs $17. How many miles did she travel?
Step 1: Choose a variable.
Let represent the number of miles traveled.
Step 2: Identify the starting amount.
The taxi charges $3 before counting any miles. This is the initial value.
Step 3: Identify the rate.
The cost increases by $2 for each mile, so the variable cost is dollars.
Step 4: Represent the total cost.
The total cost is the starting fee plus the cost per mile:
Step 5: Set the expression equal to the known total.
The total cost is $17, so write:
This equation matches the context:
The equation could also be written as a linear function:
where is the total cost. Since the problem tells us the cost is $17, we set , giving .
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