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Chapter 0: Problem 61

simplify each algebraic expression. $$ 5(3 x-2)+12 x $$

Short Answer

Expert verified
The simplified form of the given algebraic expression is \(27x - 10\).

Step by step solution

01

Distribute multiplication across the bracket

Distribute the multiplication of 5 across the term \(3x - 2\). This results in the expression \(5*3x - 5*2 + 12x\), which simplifies to \(15x - 10 + 12x\).
02

Combine like terms

Combine the terms that involve the variable \(x\) to further simplify the expression. This results in the expression \(15x + 12x - 10\), which simplifies to \(27x - 10\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distribute Multiplication
Understanding how to distribute multiplication over addition or subtraction in algebra is crucial for simplifying expressions. It involves applying the distributive property, which states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products together.

For example, if you encounter an expression like \(5(3x - 2)\), you distribute the 5 to both \(3x\) and \(2\), multiplying them separately. This results in \(15x - 10\). It's important to carry out this step correctly as it sets the stage for further simplification by combining like terms later on.
Combine Like Terms
Once you have distributed multiplication across an algebraic expression, the next step involves combining like terms to simplify the expression further. Like terms are terms that have the same variable raised to the same power. These can be combined by adding or subtracting the coefficients.

For instance, after distribution, if we have \(15x - 10 + 12x\), we notice \(15x\) and \(12x\) are like terms because they both contain the variable \(x\) raised to the first power. The coefficients can be added together, resulting in \(15x + 12x = 27x\), which simplifies the expression to \(27x - 10\).
Algebraic Simplification
The process of algebraic simplification is about reducing an expression to its most basic form without changing its value. This entails distributing multiplication, combining like terms, factoring, and canceling when possible. Each step strips away complexity, aiming to leave you with an expression that is easier to understand and work with.

For instance, our initial problem \(5(3x - 2) + 12x\), after applying distribution and combining like terms, is simplified to \(27x - 10\). The goal is to ensure that every step maintains the original expression's value while striving for the simplest form possible.
Variable Expressions
A variable expression is an algebraic statement that includes numbers, variables (like \(x\), \(y\), etc.), and operation signs. Variables stand in for unknown values and can change. The purpose of simplifying variable expressions is to make them more manageable and to clarify what the expression is equivalent to for different values of the variables involved.

For example, in \(27x - 10\), \(x\) is the variable, and the expression's value depends on what number \(x\) represents. Being efficient in simplifying variable expressions is fundamental, as it lays the foundation for solving more complex equations and understanding algebraic relationships.

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Most popular questions from this chapter

Simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. $$ \frac{y^{2}+7 y-18}{y^{2}-3 y+2} $$

Simplify each exponential expression $$ \left(\frac{3 x^{4}}{y}\right)^{-3} $$

In Exercises \(1-10\), factor out the greatest common factor. $$x^{2}(2 x+5)+17(2 x+5)$$

In Exercises \(11-16,\) factor by grouping. $$x^{3}-x^{2}+2 x-2$$

In Exercises \(11-16,\) factor by grouping. $$x^{3}-x^{2}-5 x+5$$
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