Question

In Exercises 62 and \(63,\) describe the error. $$\begin{aligned}&y=\frac{1}{3} x-2\\\&\begin{array}{l}y=3\left(\frac{1}{3} x-2\right) \\ y=x-6\end{array}\end{aligned}$$

Short answer

The error occurred in the incorrect application of the distributive property when multiplying \(3\) with \(\frac{1}{3}x-2\). Correctly, it should have been \(y = x - 6\), not \(y = x-6\).

Step-by-step solution

Identify the Error

The error in the exercise is in the second equation where \(3\) multiplies \(\frac{1}{3}x-2\). This is a wrong application of the distributive property.

Explain the Distributive Property

The distributive property is a property of real numbers that says \(a(b + c) = ab + ac\). That is, the outside number multiplies with each number inside the bracket individually.

Apply the Correct Distributive Property

Applying correctly this property to the given equation, it becomes \(y = 3*\frac{1}{3}x - 3*2\), which simplifies to \(y = x - 6\).

Key concepts

Distributive Property

One of the most fundamental principles in algebra is the distributive property. It's a rule that allows us to multiply a single term by each term in a parenthesis. The formal statement of the distributive property is:
\forall a, b, c \text{ in real numbers, } a(b + c) = ab + ac.
For example, if we have an expression like 3(2 + 4), using the distributive property, we would do 3*2 + 3*4 to get 6 + 12, which sums up to 18.

Common Mistakes

Students often apply the distributive property incorrectly by either not multiplying the outside term by each term inside the parentheses or by misapplying the operations of addition and subtraction within the parentheses. Let's rectify a common error:
If you come across an equation like \(3(\frac{1}{3}x - 2)\), the correct use of the distributive property would be to multiply 3 by both \(\frac{1}{3}x\) and -2 separately, not together as a single unit. So, the right simplification is \(3*\frac{1}{3}x - 3*2\) which simplifies to \(x - 6\), just as you follow through the multiplication and subtraction respectively.
This mistake is easily made but with practice, understanding, and attention to each component of the expression, you can avoid it.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. These expressions represent quantities that can vary, which is why we use variables like \(x\), \(y\), or \(z\) to symbolize them.
For instance, \(\frac{1}{3}x - 2\) is an algebraic expression where \(x\) is the variable. This expression states that whatever value x may take, it will be divided by 3 first, and then 2 will be subtracted from the result.

Building Blocks of Algebra

These expressions are the building blocks for forming equations and inequalities that you solve in algebra. They can be as simple as \(x + 5\) or as complex as \(\frac{2x^2 + 3xy - y^2}{x - 4}\).
Important to note is how we combine like terms (terms with the same variables raised to the same power) which simplifies the expression. When solving algebra problems or simplifying expressions, always look for like terms and use appropriate algebraic rules, such as the distributive property, to combine them.

Simplifying Equations

Simplifying equations is essentially breaking down complex expressions into the most basic form, making them easier to interpret or solve. The objective is to transform the equation into a simpler version without changing its value or solutions.
You will often achieve this by applying a series of algebraic operations such as addition, subtraction, multiplication, division, and factoring. One common method is to use the distributive property, which allows you to remove parentheses by distributing the multiplication over addition or subtraction within the parentheses.

Step-by-Step Simplification

In a multi-step problem like simplifying \(3(\frac{1}{3}x - 2)\), the first step is to identify which operations can be applied. Here, that would be the distributive property. Apply it correctly: \(3*\frac{1}{3}x\) simplifies to \(x\) and \(3*(-2)\) simplifies to \(-6\). Thus, the complex expression simplifies to \(x - 6\).
Simplification is a crucial skill because it lays the groundwork for more advanced aspects of algebra such as solving equations and inequalities. Moreover, a well-simplified equation is often easier to understand, easier to work with, and less prone to errors in further calculations.