Question

COMBINING LIKE TERMS Apply the distributive property. Then simplify by combining like terms. $$ 4(2-a)-a $$

Short answer

The simplified form of the expression \(4(2-a)-a\) is \(8-5a\)

Step-by-step solution

Apply the Distributive Property

First apply the distributive property to \(4(2-a)\) to remove the parentheses. The distributive property is expressed as \( a(b + c) = ab + ac \) . In this case, \( a = 4 \) , \( b = 2 \) , and \( c = -a \) . The result of applying the distributive property is \( 4*2 + 4*(-a) \)

Simplify the Multiplication

Next simplify the multiplication resulting from applying the distributive property. Simplify \( 4*2 \) to get \(8\), and \( 4*(-a) \) to get \(-4a\). So, now the expression is \(8-4a-a\)

Combine Like Terms

Lastly, combine the like terms, which are \(-4a\) and \(-a\). Remember that subtracting a is same as subtracting 1a, so the terms to combine are \(-4a\) and \(-1a\). This results in \(-5a\). So, the simplified expression is now \(8-5a\)

Key concepts

Distributive Property

The distributive property is a cornerstone of algebra that allows us to simplify expressions and multiply numbers in an efficient way. It states that you can distribute a single term across terms inside parentheses. For instance, if you have an expression like \( a(b + c) \), it translates to \( ab + ac \). This property is particularly helpful when you're dealing with multiplication across addition or subtraction.

Let's explore this concept with our exercise example. When we have \( 4(2 - a) \), applying the distributive property means multiplying the 4 by both terms inside the parentheses, resulting in \( 4 * 2 \) and \( 4 * (-a) \). It's like handing out the 4 to each term inside, turning the grouped expression into a simplified one without parentheses. This step is fundamental because it lays the groundwork for further simplification of algebraic expressions.

Simplify Expressions

Simplifying expressions is about making them easier to understand and work with. Think of it as tidying up a cluttered room so you can see everything clearly. After using the distributive property, as seen in our exercise, we had the expression \( 8 - 4a - a \), which still looks a bit messy. Simplifying it involves combining like terms and reducing everything to the most straightforward expression possible.

In algebra, 'like terms' are terms that have identical variable parts, like \( -4a \) and \( -a \) in our exercise. They're like apples to apples—easy to combine. To simplify \( -4a - a \), we see that \( -a \) is the same as \( -1a \) (because every number implicitly has a 1 in front of it when no other number is shown). So, combining them, we get \( -5a \). Now, our expression is neatly simplified to \( 8 - 5a \), which looks much less cluttered.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. It's like a recipe for baking a cake, where each ingredient must be mixed in a certain way to get the final product. Variables in algebraic expressions are represented by letters like \( a \), \( b \), or \( x \), and they can take on various values. Numbers that multiply variables are called coefficients.

Our exercise presented us with an algebraic expression \( 4(2 - a) - a \) that we simplified. The expression started with a mix of numbers, a variable \( a \), and operations such as multiplication and subtraction. Through the process of simplifying the expression using the distributive property and combining like terms, we ended up with the final simplified form: \( 8 - 5a \). Algebraic expressions are the building blocks of algebra and understanding how to manipulate them is essential for solving algebra problems.