Problem 31 - Algebra for College Students

Question

Simplify the algebraic expressions in Problems \(15-34\) by removing parentheses and combining similar terms. $$3(2 x-1)-4(x+2)-5(3 x+4)$$

Step-by-Step Solution

Answer

The simplified expression is \(-13x - 31\).

Step 1: Expand the first parentheses

To simplify the expression, we start by expanding each part of it. For the first part \(3(2x-1)\), distribute the \(3\) into the parentheses: \(3 \times 2x - 3 \times 1 = 6x - 3\).

Step 2: Expand the second parentheses

Next, expand the second part \(-4(x+2)\). Distribute the \(-4\) into the parentheses: \(-4 \times x - 4 \times 2 = -4x - 8\).

Step 3: Expand the third parentheses

Expand the third part \(-5(3x+4)\). Distribute the \(-5\) into the parentheses: \(-5 \times 3x - 5 \times 4 = -15x - 20\).

Step 4: Combine the expanded terms

Combine all the expanded terms into one expression: \(6x - 3 - 4x - 8 - 15x - 20\).

Step 5: Combine like terms

Identify and combine like terms from the expression: \((6x - 4x - 15x)\) for the \(x\) terms, and \((-3 - 8 - 20)\) for the constant terms. This simplifies to \(-13x - 31\).

Key Concepts

Expanding Expressions

Expanding expressions is a crucial first step in simplifying algebraic expressions. It involves removing the parentheses by distributing numbers outside of them across the terms within. For example, when we encounter an expression like \(3(2x-1)\), we need to multiply each term inside the parentheses by the 3 outside. This gives us:

• \(3 \times 2x = 6x\)
• \(3 \times -1 = -3\)

Thus, \(3(2x-1)\) becomes \(6x - 3\). Repeat this step for other parts of the expression:

• For \(-4(x+2)\), distribute the \(-4\):
• \(-4 \times x = -4x\)
• \(-4 \times 2 = -8\)

So, \(-4(x+2)\) turns into \(-4x - 8\). Continuing this way helps break down complex expressions and makes the next steps easier.
Once all parts are expanded, you can see the full expression laid out, which is crucial for systematically simplifying it.

Combining Like Terms

After expanding expressions, the next task is combining like terms. "Like terms" are terms that have the same variable parts. For instance, \(6x\), \(-4x\), and \(-15x\) from our example are all like terms because they each have the variable \(x\).To combine them, simply add or subtract their coefficients:

• \(6x - 4x = 2x\)
• \(2x - 15x = -13x\)

Similarly, combine the constant terms, which are the numbers without variables:

• \(-3 - 8 = -11\)
• \(-11 - 20 = -31\)

Now, the original expression \(3(2x-1)-4(x+2)-5(3x+4)\) simplifies to \(-13x - 31\). By carefully identifying and combining like terms, you concentrate the expression into its simplest form.

Distributive Property

The distributive property is a powerful tool in algebra. It allows you to multiply a single term by each term inside a set of parentheses. This property is defined as \(a(b + c) = ab + ac\). In the exercise, the distributive property helps in expanding expressions:

• For \(3(2x-1)\), multiply 3 with both \(2x\) and \(-1\), resulting in \(6x - 3\).
• Apply the same for \(-4(x + 2)\) and \(-5(3x + 4)\).

By the end of this property, you break larger expressions into smaller, manageable parts.
Applying the distributive property effectively lays the groundwork for further simplification steps, such as combining like terms. This approach streamlines complex problems and is an essential skill for manipulating and solving algebraic equations.