Problem 62 - Elementary Algebra

Question

\(4(3 x+5)-8(6 x-1)\)

Step-by-Step Solution

Answer

-36x + 28

Step 1 - Distribute the Multiplication

First, distribute the 4 into the term inside the parentheses: Use the distributive property: 4(3x) + 4(5) = 12x + 20.

Step 2 - Expand the Second Term

Next, distribute the -8 into the second set of parentheses: Use the distributive property again: -8(6x) - 8(-1) = -48x + 8.

Step 3 - Combine Like Terms

Now, combine the results of the distributions from the previous steps: (12x + 20) + (-48x + 8). To simplify, combine the like terms: (12x - 48x) + (20 + 8) = -36x + 28.

Key Concepts

Combining Like Terms

In algebraic expressions, 'like terms' are terms that have the same variable raised to the same power. For instance, in the expression \(12x - 48x\), both terms are like terms because they contain the variable \(x\) raised to the first power. To combine like terms, we simply add or subtract their coefficients. Here’s a simple breakdown:

• If you have \(3x + 4x\), you combine by adding the coefficients \(3 + 4\) to get \(7x\).
• If you have \(5y - 2y\), subtract the coefficients \(5 - 2\) to get \(3y\).

From our exercise, combining \(12x - 48x\) involves subtracting coefficients: \(12 - 48 = -36\). Hence, \(12x - 48x = -36x\). It’s just simple arithmetic!

Algebraic Expressions

An algebraic expression includes numbers, variables, and arithmetic operations. There’s no equal sign in an algebraic expression, unlike in an equation. Examples of algebraic expressions include:

• \(4x + 7\)
• \(3y - 5 + 2z\)
• \(12x + 5 - 3\)

The given exercise \(4(3x+5)-8(6x-1)\) is an algebraic expression that we need to simplify. In this expression, parentheses indicate which terms should be dealt with together first using the distributive property. Once we get rid of the parentheses, we combine like terms to simplify further.

Simplification

Simplification involves reducing an algebraic expression to its simplest form. This usually involves removing parentheses, combining like terms, and performing any necessary arithmetic operations. For our exercise:

• First, we expand using the distributive property: \(4(3x) + 4(5) - 8(6x) - 8(-1) = 12x + 20 - 48x + 8\).
• Next, we combine like terms: \(12x - 48x + 20 + 8\).
• This gives us: \(-36x + 28\).

Remember, the goal of simplification is to make the expression as straightforward and concise as possible, removing any unnecessary complexity.