Question

1. real-world modeling: a taxi charges a flat fee of \\$5.00 plus \\$2.50 per mile. if the total fare was \\$20.00, write and solve an equation to find the number of miles ((m)) traveled.

• equation: _______
• solution: (m =) _______

1. error analysis: a student solved the equation (2(x - 4) = 16) and got (x = 10). are they correct? show your work to justify your answer.

Explanation:

Set up the linear model

Using the Linear Modeling knowledge point:
Let \(m\) represent the number of miles traveled.
The flat fee is \(\$5.00\).
The rate per mile is \(\$2.50\).
The total fare is \(\$20.00\).
The equation is:

$$5.00 + 2.50m = 20.00$$

Solve for the number of miles

Using the Solving Multi-Step Equations knowledge point:
Subtract \(5.00\) from both sides:

$$2.50m = 15.00$$

Divide both sides by \(2.50\):

$$m = 6$$

Analyze the student's solution

Using the Algebraic Error Analysis knowledge point:
The student solved:

$$2(x - 4) = 16$$

Let's solve this equation step-by-step to verify:
Divide both sides by \(2\):

$$x - 4 = 8$$

Add \(4\) to both sides:

$$x = 12$$

Identify the student's error

The student obtained \(x = 10\).
If we substitute \(x = 10\) back into the original equation:

$$2(10 - 4) = 2(6) = 12 eq 16$$

Thus, the student's answer is incorrect.
The student likely added \(4\) to \(16\) first, or made an arithmetic error during distribution.

Answer:

Question 5

• Equation: \(5.00 + 2.50m = 20.00\)
• Solution: \(m = 6\)

Question 6

No, the student is incorrect.

Solving the equation correctly:

$$2(x - 4) = 16$$

$$x - 4 = 8$$

$$x = 12$$

If we substitute the student's answer \(x = 10\) back into the equation:

$$2(10 - 4) = 2(6) = 12 eq 16$$