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Chapter 1: Problem 94

Simplify. $$ 2 a-(5 a-9) $$

Short Answer

Expert verified
-3a + 9.

Step by step solution

01

Distribute the Negative Sign

Inside the parentheses, distribute the negative sign to both terms. This changes the expression as follows: 2a - (5a - 9) becomes 2a - 5a + 9.
02

Combine Like Terms

Combine the like terms, which are the terms involving 'a'. 2a - 5a simplifies to -3a.
03

Write the Final Expression

Add the remaining constant term: -3a + 9.

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Key Concepts

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

Distributing Negative Signs
When simplifying algebraic expressions, the first step often involves distributing negative signs. This means you need to apply the negative sign to each of the terms inside the parentheses.

For example, in the expression \( 2a - (5a - 9) \), the negative sign in front of the parentheses affects both \( 5a \) and \( -9 \).

Think of the negative sign as multiplying each term inside the parentheses by -1. So, we can rewrite \( -(5a - 9) \) as \( -5a + 9 \).
  • Minus 5a becomes negative 5a.
  • Minus a negative 9 becomes positive 9, as subtracting a negative is like adding a positive.
Now the expression is transformed to \( 2a - 5a + 9 \). This step is crucial for correctly simplifying expressions, as mistakes here can lead to wrong results.
Combining Like Terms
Once you've distributed any negative signs, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power.

In our example, after distributing the negative sign, we get \( 2a - 5a + 9 \). Here, \( 2a \) and \( -5a \) are like terms because they both involve the variable 'a'.

To combine these, simply add or subtract the coefficients (the numbers in front of the variables). For \( 2a - 5a \):
  • Take the coefficient of \(2a\), which is 2.
  • Subtract the coefficient of \(5a\), which is 5.
This gives us \(2 - 5 = -3\). So, \( 2a - 5a \) simplifies to \( -3a \). Be sure to keep the variable part the same.

After combining the like terms, our expression now is \( -3a + 9 \).
Algebraic Simplification
The final step in simplifying algebraic expressions is to make sure that all like terms are combined and that the expression is in its simplest form. In our example, we have reached \( -3a + 9 \).

This means our simplified expression contains a single variable term and a single constant term.
  • If there were any further like terms, repeat the process of combining them.
  • Always double-check your distribution steps.
More complex expressions might require additional steps like factoring or using the distributive property in different ways, but the key principles remain:
  • Distribute signs correctly.
  • Combine like terms accurately.
  • Simplify step-by-step.
With practice, these steps will become second nature. Understanding and practicing these basic steps will help you simplify even the most complex algebraic expressions with confidence.

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