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Chapter 6: Problem 42

Simplify \(a \div(5 a+2 a)-3 a\)

Short Answer

Expert verified
The simplified expression is \(\frac{1}{7} - 3a\).

Step by step solution

01

Simplify the Division Inside the Parentheses

First, simplify the term inside the parentheses: \[5a + 2a.\]Combine like terms:\[5a + 2a = 7a.\] So, the expression becomes \[a \div 7a - 3a.\]
02

Simplify the Division

Now, focus on the division operation. The expression \[a \div 7a\]can be expressed as:\[\frac{a}{7a}.\]Simplify this by canceling out 'a' from the numerator and the denominator:\[\frac{1}{7}.\]Now the expression is:\[\frac{1}{7} - 3a.\]
03

Combine the Terms

Finally, simplify the overall expression. Since there are no like terms (\(\frac{1}{7}\) and \(-3a\) are not like terms), the simplified expression is:\[\frac{1}{7} - 3a.\]

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Key Concepts

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

Combining Like Terms
Combining like terms is a key skill in simplifying algebraic expressions. When you look at an expression, like \(5a + 2a\), you should first identify terms that have the same variables raised to the same powers. In this case, both terms are 'like' because they share the variable 'a'.
  • Terms are considered like if they have the same variables and the same exponents.
  • Combining these like terms involves simply adding the coefficients, which are the numbers multiplying the variables.
  • So, \(5a + 2a\) becomes \(7a\).
By reducing terms like these, you simplify the expression, making it easier to handle in further operations. It鈥檚 one of the initial steps to clean up an algebraic expression before diving into more complex operations like division or factoring.
Division in Algebra
When you encounter division in algebra, remember that it often involves rewriting the expression using fractions. In the exercise, you see \(a \div 7a\), which can be represented as \(\frac{a}{7a}\). This transformation is helpful because it sets the stage for further simplification.

Here's how the division works:
  • Turn the division notation into a fraction. This means \(a \div 7a\) becomes \(\frac{a}{7a}\).
  • Fractions can often be simplified by canceling common factors, especially when the numerator and the denominator have similar terms.
Keep in mind that algebraic division requires attention to detail. Ensuring that you handle each component of the expression correctly will result in a much simpler form.
Canceling Common Factors
Canceling common factors in algebraic fractions is akin to simplifying numerical fractions. You've probably heard of reducing fractions before. It works the same way in algebra.

In the fraction \(\frac{a}{7a}\), note that both the numerator and the denominator include the variable 'a'. This is the common factor you can remove.
  • Strike out or "cancel" the 'a' in both the numerator and the denominator to simplify the fraction.
  • This leaves you with the simplified version of the fraction: \(\frac{1}{7}\).
Canceling common factors helps simplify expressions greatly because it reduces the complexity before combining or subtracting other terms. Always check for these common factors when dealing with fractions in algebra to keep things as simple as possible.

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