Problem 32 - Elementary Algebra

Question

Simplify. (Assume all denominators are nonzero.) $$ 3 x 13-1 x $$

Step-by-Step Solution

Answer

The expression is simplified to 39 - x.

Step 1: Distribution

First, distribute the numbers outside the parentheses into the terms inside. Here, it involves multiplying the coefficient 3 by the expression it affects. However, in this particular problem, it appears a distribution of multiplication is unnecessary due to a formatting typo, and we will simplify erroneously thought brackets directly into a subtraction process instead.

Step 2: Simplify multiplication

Next, observe each term and ensure they are in their simplest form. For the number present, confirm if there is any multiplication that needs to be resolved. In this case, it appears `3x13` and then adjust subtraction '1x', simply rewriting the expression correctly.

Step 3: Perform multiplication

Calculate the multiplication of numbers suspected: \[3 \times 13 = 39\]Therefore, rewrite the expression as:\[39 - 1x\]

Step 4: Combine like terms

Finally, combine any like terms if possible. Look for any x terms that can be further simplified. Since there is an "x" term here that stands alone, we leave it as is:\[39 - x\]

Key Concepts

Distributive Property

The distributive property is a key algebraic property used to simplify expressions. It states that when you multiply a number by a sum, you can distribute the multiplication over the terms inside the parentheses. In math, it's expressed as:

• \(a(b + c) = ab + ac\)

In the context of the exercise \(3x13 - 1x,\)there was no need for the distributive property because the expression formatting was incorrect.
This property is often used when simplifying expressions with more complex structures. But sometimes, as in this exercise, a direct multiplication is enough to simplify.
Understanding when and how to apply this property helps avoid unnecessary steps and makes simplification quicker.

Like Terms

Like terms are terms that contain the same variable raised to the same power. In the expression \(39 - x,\)there is only one term with a variable, which is \(-x\).When simplifying expressions, combining like terms means grouping them together.

• Example: In \(2x + 3x = 5x,\)the terms \(2x\)and \(3x\)are like terms because they both contain the variable \(x\).

In our problem, since there's only one term with \(x\),no further combining is necessary.
This concept is crucial for making expressions cleaner and simpler, especially when dealing with longer algebraic problems.

Multiplication in Algebra

Multiplication in algebra works similarly to arithmetic multiplication but involves variables alongside numbers.
It requires multiplying coefficients (the numerical part) together while maintaining the variables intact until combining like terms.
In our exercise, \(3 \times 13\)was calculated first as part of simplifying the expression. When multiplying a number by a variable, simply place them together:

• Example: \(2 \times x = 2x\)

For more complex expressions, use the order of operations rules. Handle multiplication before addition and subtraction.
Recognizing how to perform multiplication efficiently is vital for success in algebra. It helps in simplifying expressions quickly and accurately.