Model real-world situations using linear equations

This understanding involves translating a real-world relationship between quantities into a linear equation, identifying the variable, fixed value, rate of change, and units, and interpreting forms such as $y=mx+b$ in context. Learners connect tables, graphs, verbal descriptions, and equations, solve for an unknown, and judge whether the solution is meaningful within the situation’s constraints; systems, piecewise relationships, and nonlinear models are beyond this scope.

Detailed Explanation: Model real-world situations using linear equations

A linear equation shows how one quantity changes at a constant rate from a starting value.

In the form

y=mx+b

$y$ is the total or changing quantity.
$x$ is the input or variable.
$m$ is the rate of change.
$b$ is the fixed starting value.

Example

A taxi charges a fixed fee of $4 plus $2.50 for each mile traveled. How many miles can you travel for $19?

1. Choose the variables.

Let

$x=$ number of miles traveled
$y=$ total cost in dollars

2. Identify the fixed value and rate.

The fixed fee is $4, so $b=4$ .
The cost increases by $2.50 for every mile, so $m=2.50$ dollars per mile.

3. Write the equation.

Using $y=mx+b$ :

y=2.50x+4

This equation means the total cost is $2.50 for each mile, plus the $4 starting fee.

4. Use the given total cost.

The trip costs $19, so substitute $19$ for $y$ :

19=2.50x+4

5. Solve for $x$ .

Subtract $4$ from both sides:

15=2.50x

Divide by $2.50$ :

x=6

6. Interpret the answer.

You can travel 6 miles for $19. The answer is meaningful because a distance cannot be negative, and $6$ miles fits the situation.

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