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

We define the variable \(m\) as the number of miles traveled. The flat fee is \(\$5.00\) and the cost per mile is \(\$2.50\). The total fare is \(\$20.00\). Using Linear Equations, we write:

$$2.50m + 5.00 = 20.00$$

Solve for the number of miles

Using Solving Multi-Step Equations, we isolate \(m\):

$$2.50m = 15.00$$

$$m = \frac{15.00}{2.50} = 6$$

Analyze the student's solution

The student solved \(2(x - 4) = 16\) and got \(x = 10\). We substitute \(x = 10\) back into the original equation to check for correctness:

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

Since \(12
eq 16\), the student's answer is incorrect.

Solve the equation to justify

We solve the equation correctly to show the correct value:

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

$$x - 4 = 8$$

$$x = 12$$

The correct solution is \(x = 12\).

Answer:

Question 5

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

Question 6

No, they are not correct.

Justification:
If we substitute \(x = 10\) into the equation:

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

Solving the equation correctly:

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

$$x - 4 = 8$$

$$x = 12$$

The correct solution is \(x = 12\).