Problem 56 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 8 x-3 x+4(2 x+5)+3(6 x-4) $$

Step-by-Step Solution

Answer

Question: Simplify the following algebraic expression: $$4(2x + 5) - 3(6x - 4)$$. Answer: The simplified algebraic expression is $$31x + 8$$.

Step 1: Distribute numbers inside parentheses

To simplify the expression, we must first distribute the numbers inside the parentheses to the terms inside. We have two such cases here: $$4(2x + 5)$$ and $$3(6x - 4)$$. Distributing the numbers inside these parentheses, we get: $$8x - 3x + 8x + 20 + 18x -12$$.

Step 2: Combine like terms

Now that we have distributed the numbers, we can combine like terms. We are looking for terms that involve the same variable raised to the same power. In this case, we have three terms containing x:$$8x$$, $$-3x$$, $$8x$$ and $$18x$$. Combining these terms, we get: $$8x - 3x + 8x + 18x = 31x$$. We also have two constant terms: $$20$$ and $$-12$$. Combining these constant terms gives: $$20 - 12 = 8$$.

Step 3: Write the simplified expression

After distributing the numbers and combining like terms, we are left with $$31x$$ and $$8$$. Combining these, we get the simplified algebraic expression: $$31x + 8$$.

Key Concepts

Distributive Property

The distributive property is a core concept in algebra that helps simplify expressions and make calculations easier. It allows you to multiply a single term across terms inside a bracket, effectively removing the parentheses. This property is expressed as \( a(b + c) = ab + ac \).

Applying this rule to our exercise, we started with two distributions:

• \( 4(2x+5) \), which means multiplying 4 across both 2 and 5 inside the parentheses.
• \( 3(6x-4) \), which means multiplying 3 across both 6x and -4.

Once distributed, both expressions lose the brackets and simplify to a sum of separate terms:

• \( 4(2x+5) = 8x + 20 \)
• \( 3(6x-4) = 18x - 12 \)

Practicing the distributive property helps reinforce basic multiplication skills and paves the way for solving more complex algebraic problems.

Combining Like Terms

In algebra, combining like terms is crucial for simplifying expressions. "Like terms" refer to terms that have the same variable raised to the same power. They can be collected together through addition or subtraction.

In our exercise, after applying the distributive property, we ended up with several terms involving \( x \) and constants:

• Terms with \( x \): \( 8x, -3x, 8x, \) and \( 18x \)
• Constant terms: \( 20 \) and \( -12 \)

To combine like terms, simply add or subtract the coefficients of terms with \( x \): \( 8x - 3x + 8x + 18x = 31x \). Similarly, combine constants: \( 20 - 12 = 8 \).

This step streamlines your expression, making it easier to interpret or solve. Recognizing and combining like terms is a fundamental skill in algebra.

Algebraic Expressions

Algebraic expressions form the backbone of algebraic thinking. They are made up of numbers, variables, and mathematical operations like addition, subtraction, multiplication, and division. Expression simplification is a process involving different algebraic techniques, like using the distributive property and combining like terms.

In the original problem, the expression: \( 8x - 3x + 4(2x+5) + 3(6x-4) \), includes terms with variables and constants. Our goal was to simplify it using these techniques, resulting in the streamlined form \( 31x + 8 \).

Simplifying algebraic expressions helps you solve equations, understand algebra-related problems, and reveal the essential structure of any mathematical problem. Mastery of these skills will be invaluable for more advanced math topics and real-world applications.