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Chapter 11: Problem 70

Collect like terms. $$ 6.7 a+4.3 b-4.1 a-2.9 b $$

Short Answer

Expert verified
The simplified expression is \(2.6a + 1.4b\).

Step by step solution

01

Identify Like Terms

First, identify the terms in the given expression that have the same variables. The given expression is: \(6.7a + 4.3b - 4.1a - 2.9b\). The like terms here are \(6.7a\) and \(-4.1a\), and \(4.3b\) and \(-2.9b\).
02

Combine Like Terms for \(a\)

Combine the terms that have the variable \(a\): \(6.7a - 4.1a\). Subtract \(4.1\) from \(6.7\) to get \(2.6a\).
03

Combine Like Terms for \(b\)

Next, combine the terms that have the variable \(b\): \(4.3b - 2.9b\). Subtract \(2.9\) from \(4.3\) to get \(1.4b\).
04

Write the Simplified Expression

Finally, write the simplified expression by combining the results from the steps above: \(2.6a + 1.4b\).

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Key Concepts

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

Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations (like addition and subtraction). They don't have an equality sign, so they are different from equations. For example, in the given problem, the expression is: \(6.7a + 4.3b - 4.1a - 2.9b\). Here, \(a\) and \(b\) are variables, while the numbers (6.7, 4.3, 4.1, 2.9) are coefficients. Algebraic expressions can be simplified by combining like terms. This makes them easier to work with and understand. Simplifying helps in solving more complex algebraic problems.
Like Terms
Like terms are terms in an algebraic expression that have the same variables raised to the same power. They can be combined to simplify the expression. In our problem, the like terms are \(6.7a\) and \(-4.1a\), and \(4.3b\) and \(-2.9b\). The variables in like terms must exactly match. So, you can add or subtract their coefficients, but the variable part stays the same. Combining like terms makes an expression shorter and easier to manage.
Simplifying Expressions
Simplifying algebraic expressions means reducing them to their simplest form. One way to do this is by combining like terms. Begin by identifying like terms, then combine their coefficients. Use basic arithmetic to add or subtract these coefficients.
In our example:
  • Identify \(6.7a\) and \(-4.1a\) as like terms.
  • Subtract \(4.1\) from \(6.7\) to get \(2.6a\).
  • Then identify \(4.3b\) and \(-2.9b\) as like terms.
  • Subtract \(2.9\) from \(4.3\) to get \(1.4b\).
The simplified expression is \(2.6a + 1.4b\).
Variables
Variables are symbols (usually letters like \(a\), \(b\), \(x\), \(y\)) used to represent unknown numbers. They are essential in algebra because they allow us to write general mathematical expressions and equations. In the given expression \(6.7a + 4.3b - 4.1a - 2.9b\), \(a\) and \(b\) are variables. They can stand for any number, and the expression shows relationships between these numbers. Variables make it possible to generalize problems and find solutions that work for many different cases. Understanding how to manipulate variables is a key skill in algebra.

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