Problem 109 - College Algebra Essentials

Question

Suppose you are an algebra teacher grading the following solution on an examination: $$\begin{aligned}-3(x-6) &=2-x \\\\-3 x-18 &=2-x \\\\-2 x-18 &=2 \\\\-2 x &=-16 \\\x &=8\end{aligned}$$ You should note that 8 checks, so the solution set is \(\\{8\\}\) The student who worked the problem therefore wants full credit. Can you find any errors in the solution? If full credit is 10 points, how many points should you give the student? Justify your position.

Step-by-Step Solution

Answer

The student made a mistake in the distribution of the term -3(x - 6) where they wrote -18 instead of +18. Nonetheless, the student managed to get the correct answer although using incorrect steps. Out of 10 points, considering the mistake was not fundamental and the final result was correct, a fair score could be 8 points.

Step 1: Identify the error on the student's solution

Looking at the steps provided by the student, it can be seen that a mistake has been made while distributing -3 in the first step. -3 multiplied by -6 should have resulted in +18, not -18.

Step 2: Evaluate the correct solution

The correct series of steps should go as follows: -3(x - 6) equates to -3x + 18. After this, when we equate -3x + 18 = 2 - x, by combining like terms, we have -2x + 18 = 2. Extracting 18 from both sides of the equation, we get -2x = -16, so x = 8 after dividing both sides by -2. This confirms that the derived solution by the student was correct, despite the wrong step.

Step 3: Check the solution

Substituting x = 8 into the original equation to verify -3(8 - 6) = 2 - 8, -6 does indeed equal -6, confirming the solution's validity.

Step 4: Decide on the score given the error

Despite getting the final solution correct, the student made a mistake in their working which could lead to a wrong solution in other circumstances. Therefore, they do not deserve full credit.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term across a sum or difference. In this context, it involves the multiplication of a number by each term inside a set of parentheses. For example, if we have the expression \(-3(x - 6)\), the distributive property tells us to multiply \(-3\) by both \(x\) and \(-6\), resulting in \(-3x + 18\).
This step is crucial because overlooking it can lead to incorrect solutions later on. In the original problem, the student made an error by incorrectly multiplying \(-3\) by \(-6\), which highlights the importance of careful arithmetic during distribution.

• Ensure that each step is double-checked.
• Be mindful of the signs (+ or -) before distributing.

Understanding and correctly applying the distributive property is essential for solving equations accurately.

Equation Solving

Solving equations is a core activity in algebra, where the goal is to find the value of the unknown variable that makes the equation true. In this exercise, equation solving involves several steps:
1. Use the distributive property to simplify the equation.
2. Combine like terms on each side of the equation.
3. Move terms to isolate the variable on one side.
4. Solve for the variable.
In this particular example, the student correctly identified the final value of \(x\) to be \(8\) despite an initial error with the distributive property. This highlights an important lesson: even if a step seems accurate in execution, each part of the process needs to be continuously verified.
Checking your solution is always a good practice. Substitute your answer back into the original equation to ensure both sides equal up.

Student Solutions

Student solutions can be riddled with small errors that might seem insignificant but can lead to larger issues in different scenarios. In the reviewed example, although the student landed on the correct answer, there was an error in the algebraic manipulation due to an incorrect application of the distributive property.
This emphasizes the value of looking at solutions holistically. Understanding where mistakes occur can help prevent them in future problems.

• Review each step thoroughly before concluding.
• Understand the rationale behind each algebraic manipulation.

Encourage students to reflect on their methodologies to gain a deeper understanding of each step.

Grading Criteria

When it comes to grading student work, a teacher must decide how to weigh each component of the solution process. It’s not just about the final answer; the approach and accuracy in intermediate steps matter too. In grading this student's solution:
The error in applying the distributive property is significant because it is foundational to the problem-solving process. If not caught, it could form bad habits that leave students vulnerable in exams.

• Consider both the correctness and the understanding demonstrated in the process.
• Partial credit can be given for the right approach even with missteps, but continued errors can’t be overlooked.

In this example, the student should be awarded most of the points, but a deduction should be made for the initial distribution error, perhaps scoring an 8 out of 10.