Problem 32 - Algebra for College Students

Question

Simplify the algebraic expressions in Problems \(15-34\) by removing parentheses and combining similar terms. $$-2(x-1)-5(2 x+1)+4(2 x-7)$$

Step-by-Step Solution

Answer

The simplified expression is \(-4x - 31\).

Step 1 - Distribute the coefficients in each term

First, apply the distributive property to remove the parentheses. Multiply each term inside the parentheses by the coefficient in front of the parentheses:\(-2(x-1) = -2x + 2\)\(-5(2x+1) = -10x - 5\)\(4(2x-7) = 8x - 28\)

Step 2 - Write the expanded expression

Combine the results from Step 1 into a single expression:\(-2x + 2 - 10x - 5 + 8x - 28\)

Step 3 - Combine like terms

Identify and combine like terms in the expression:Combine the \(x\) terms: \(-2x - 10x + 8x = -4x\)Combine the constant terms:\(2 - 5 - 28 = -31\)So, the expression is simplified to:\(-4x - 31\)

Key Concepts

Distributive Property

The distributive property is a crucial tool in algebra that helps simplify expressions by getting rid of parentheses. It states that when you have a number or coefficient outside of a parenthesis, you distribute or multiply it by each term inside the parenthesis. For example, if you have something like

• \(-2(x-1)\)

this expression uses the distributive property to become

• \(-2(x) - 2(1) = -2x + 2\).

This allows us to turn more complex expressions into simpler ones.
Once we use the distributive property on all sets of parentheses, we can look at the expression without those grouping symbols, ready for further simplification. This process is essential in solving or simplifying any algebraic expression.

Like Terms

When dealing with algebraic expressions, like terms are ones that have the exact same variable and the same exponent. They can be combined to simplify the expression further. Consider the expression:

• \(-2x - 10x + 8x\).

All of these are like terms because they have the variable \(x\).

• Combining them means adding or subtracting the coefficients while keeping the \(x\) variable intact:

i.e.

• \(-2x - 10x + 8x = -4x\).

It's important to spot like terms because they tell us how we can combine parts of the expression to make it simpler.
This is similar in method for constant numbers. They are also "like terms" among themselves, even if they don't have a variable attached.
Simplifying expressions using like terms makes them much easier to understand and work with.

Algebraic Expressions

In algebra, an algebraic expression is a mathematical phrase that includes numbers, variables, and operations. These expressions can sometimes appear complicated, but the process of simplifying them makes them manageable. The expression from our exercise is a perfect example:

• \(-2x + 2 - 10x - 5 + 8x - 28\).

At first glance, this might seem daunting, but following steps like using the distributive property and combining like terms helps us tackle it systematically.

• First, distribute to remove parentheses.
• Then, pair up and simplify like terms.

The end result is a much simpler form:

• \(-4x - 31\).

Algebraic expressions are foundational in algebra, used in solving equations, modeling real-world situations, and more.
Simplifying them is a skill that becomes more intuitive with practice, helping students gain confidence in handling elaborate mathematical ideas.