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Chapter 1: Problem 24

Simplify the given algebraic expressions. $$3(a-b)-2(a-2 b)$$

Short Answer

Expert verified
The simplified expression is \(a + b\).

Step by step solution

01

Distribute Constants

Expand each term in the expression by distributing the constants: The expression is \(3(a-b) - 2(a-2b)\).- Distribute \(3\) into \(a - b\) to get \(3a - 3b\).- Distribute \(-2\) into \(a - 2b\) to get \(-2a + 4b\). This gives us the expression \(3a - 3b - 2a + 4b\).
02

Combine Like Terms

Simplify the expression by combining like terms:- Combine \(3a\) and \(-2a\) to get \(a\).- Combine \(-3b\) and \(4b\) to get \(b\).Thus, the expression simplifies to \(a + b\).

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Key Concepts

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

Distributive Property
The distributive property is a fundamental concept in algebra that helps us simplify expressions by spreading out a multiplication over addition or subtraction inside parentheses. Essentially, it allows you to "distribute" a factor across terms inside a bracket.

The general formula is:
  • For any numbers or expressions, if you have something like \(a(b + c)\), it simplifies to \(ab + ac\).
  • Similarly, \(a(b - c)\) simplifies to \(ab - ac\).

In our exercise, we applied the distributive property to \(3(a-b)\) and \(-2(a-2b)\).
By distributing, we got \(3a - 3b\) from the former and \(-2a + 4b\) from the latter. This step can transform a complex expression into something more straightforward, setting the stage for the next simplification steps.
Combining Like Terms
Once distribution is complete, combining like terms is the next crucial step in simplification. Like terms have the same variable part raised to the same power, even if their coefficients are different. This means you can add or subtract these terms as you would ordinary numbers.

For example, you can combine \(5x\) and \(-3x\) into \(2x\), but you cannot combine \(5x\) and \(4y\) because they have different variables.

In the expression from our exercise, we reduced \(3a - 3b - 2a + 4b\) by recognizing the like terms:
  • Combine \(3a\) and \(-2a\), which results in \(a\).
  • Likewise, combine \(-3b\) and \(4b\) to obtain \(b\).

This process of combining yields \(a + b\), a far simpler expression that highlights the power of correct algebraic manipulation.
Algebraic Expressions
Algebraic expressions form the core foundation of algebra and incorporate numbers, variables, and arithmetic operators. Simplifying these expressions is all about making them as concise as possible while retaining their original value.

An algebraic expression like \(3(a-b)-2(a-2b)\) can initially appear complex. However, the key to handling such expressions involves steps like applying the distributive property and combining like terms, as we've seen.

Understanding algebraic expressions is crucial because they
  • provide a way to generalize mathematical problems,
  • allow for the formulation of equations and inequalities,
  • and develop critical thinking and problem-solving skills.

With practice, the art of simplifying algebraic expressions becomes intuitive, enabling you to tackle even more challenging equations effortlessly.

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Most popular questions from this chapter

Perform the indicated calculations.A TV signal travels \(1.86 \times 10^{5} \mathrm{mi} / \mathrm{s}\) for \(4.57 \times 10^{4} \mathrm{mi}\) from the station transmitter to a satellite and then to a receiver dish. How long does it take the signal to go from the transmitter to the dish?

Solve the given equations. $$|x|-1=8$$

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Solve the given equations. All numbers are approximate. $$27.5(5.17-1.44 x)=73.4$$
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