Problem 97 - Introductory and Intermediate Algebra for College Students 4th

Question

Suppose you are an algebra teacher grading the following solution on an examination: $$\begin{aligned}\text { Solve: } &-3(x-6) &=2-x \\\\\text { Solution: } &-3 x-18 &=2-x \\\&-2 x-18 &=2 \\\\-2 x &=&-16 \\\x &=8\end{aligned}$$ You should note that 8 checks, and the solution 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

Incorrect algebraic manipulations were made during the solution process for the exercise. Therefore, the student's solution doesn't warrant full marks, considering it's crucial to know the algebraic principles. A fair score would be 5 out of 10.

Step 1: Error Identification 1

Looking at the first step, the student went from \(-3(x-6)\) to \(-3x -18\). This is incorrect as according to the distributive law of multiplication over subtraction, \( -3(x - 6) \) should be \(-3x + 18\), not \(-3x -18\)

Step 2: Error Identification 2

In the second step, the student went from \(-3x -18 = 2 - x\) to \(-2x -18 = 2\). This is incorrect. When you add \(x\) to both sides to remove \(x\) from the right, the equation becomes \(-2x + 18 = 2\).

Step 3: Scoring the Solution

Considering these errors, the student's solution contains significant mistakes in the basic principles of algebra, which leads to the wrong final answer. Therefore, full credit should not be granted. As it appears the student understands the general process but made simple arithmetic errors, a score of 5 out of 10 can be considered.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra used to simplify expressions and solve equations. It deals with the distribution of a factor across the terms inside a parenthesis. For our case, the distributive property states that for any numbers or expressions \( a, b, \, \text{and} \, c \), \( a(b + c) = ab + ac \) and \( a(b - c) = ab - ac \). This means we multiply each term inside the parenthesis by the factor outside the parenthesis.
For example, when given \(-3(x - 6)\), the correct application of the distributive property would be:

• Multiply \(-3\) by \(x\) to get \(-3x\).
• Multiply \(-3\) by \(-6\) to get \(+18\).

This results in the expression \(-3x + 18\). It's crucial to remember that the sign in front of each term must be carefully considered to avoid errors. This step is vital for setting up equations correctly and is often where mistakes occur if details are overlooked.

Equation Solving

Solving an equation means finding the value of the variable that makes the equation true. This process usually involves simplifying expressions, isolating the variable, and checking the solution. Let's dive a bit deeper into the steps:
1. **Simplify Both Sides**: Before attempting to isolate the variable, simplify expressions on both sides using fundamental algebraic operations like distributing, combining like terms, or simplifying fractions.
In the example given, simplifying correctly means first using the distributive property accurately.
2. **Isolate the Variable**: Use inverse operations to get the variable alone on one side of the equation. This usually involves steps like adding, subtracting, multiplying, or dividing both sides to eliminate other terms.
3. **Check Your Solution**: Always plug your solution back into the original equation to ensure that it satisfies the equation.
That procedure ensures a comprehensive understanding and helps detect any calculation errors early. The given task can also demonstrate where the understanding might have faltered, as reflected in arithmetic errors.

Arithmetic Errors

Arithmetic errors are mistakes that occur from incorrect calculations, whether due to oversight or misunderstanding. They might seem trivial but can lead to erroneous solutions. Circumstances under which these errors often happen include:

• Misapplication of basic operations like addition or subtraction.
• Incorrect signs, especially when distributing negatives.

In the example solution, the error arises primarily around the misapplication of signs in the distributive step (-3 going to -18 instead of +18). This small mistake can propagate, leading to a correct procedural approach but a fundamentally wrong answer.
To minimize these errors, it's helpful to:

• Double-check results at each step.
• Pay attention to negative signs and ensure they are handled carefully.
• Re-evaluate calculations by substituting back into the original equation, verifying the solution satisfies it.

Being diligent about each step of the calculation process is key to yielding correct solutions without arithmetic errors.