Problem 58 - Algebra A Combined Function

Question

Remove parentheses and simplify each expression. $$ \frac{1}{3}(9 x-6)-(x-2) $$

Step-by-Step Solution

Answer

The simplified expression is \( 2x \).

Step 1: Distribute the Fraction

Multiply each term inside the parentheses by the fraction \( \frac{1}{3} \). This gives \( \frac{1}{3} \times 9x - \frac{1}{3} \times 6 = 3x - 2 \).

Step 2: Rewrite the Expression

Replace the expression from the first step into the main equation: \( 3x - 2 + (-x + 2) \). The negative sign in front of the second parenthetical flips the signs of the terms inside.

Step 3: Combine Like Terms

Look at the expression and sum up the like terms. Combine the terms with \( x \): \( 3x - x = 2x \) and combine the constant terms: \( -2 + 2 = 0 \).

Step 4: Simplify

Now, combine the results from step 3. The expression simplifies to \( 2x \).

Key Concepts

Distributive Property

The distributive property is a key concept in algebra that allows us to multiply a single term by terms inside a set of parentheses. It's like distributing a gift to each member of a group.
In a mathematical expression, such as the one you find in this exercise, the distributive property enables you to remove parentheses by distributing the multiplier to each term within. For example, when applying the distributive property to the expression \( \frac{1}{3}(9x - 6) \), you multiply each term inside the parentheses by \( \frac{1}{3} \).
This results in \( \frac{1}{3} \times 9x = 3x \) and \( \frac{1}{3} \times -6 = -2 \). Once done, the expression becomes \( 3x - 2 \), effectively distributing the fraction to each component within the parentheses.

• Remember, every term inside the parentheses gets multiplied separately.
• The distributive property also works with subtraction, where you multiply the subtracted terms as well.

Combining Like Terms

Once you've distributed or simplified an expression, the next step is to combine like terms. But what does this mean? Like terms are elements in an equation that have the same variables raised to the same power. For a smooth simplification, you sum up or subtract these similar terms.
In our exercise, after distributing the fraction, the expression was \( 3x - 2 + (-x + 2) \). To combine like terms, start by identifying terms with \( x \): \( 3x - x \).
Likewise, you combine the constant numbers: \( -2 + 2 \).

• Terms with the same base and exponent are combined by adding or subtracting their coefficients.
• Constants, numbers without variables, are also combined to streamline the expression.

Two important things happen here: The \( 3x \) and \( -x \) add to form \( 2x \), and \( -2 + 2 \) cancel each other out to become \( 0 \).

Linear Expressions

Linear expressions are algebraic expressions where each term is either a constant number or a product of a constant and a single variable. These expressions represent straight lines when graphed. They're straightforward, as they don't include exponents other than one or include variables multiplied together.
After applying the distributive property and combining like terms in the exercise, we were left with the expression \( 2x \). This is a classic example of a linear expression.

• Linear expressions do not contain variables raised to powers higher than one.
• They are the simplest form of algebraic expressions, often foundational for understanding more complex equations.

The main characteristic of a linear expression is that it contains no higher powers of \( x \), which means it will create a straight line when graphed on an x-y coordinate plane.