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Chapter 1: Problem 3

Simplify the algebraic expressions by combining similar terms. $$ 5 a^{2}-6 a^{2} $$

Short Answer

Expert verified
The simplified expression is \(-a^2\).

Step by step solution

01

Identify like terms

In the expression \(5a^2 - 6a^2\), identify the like terms. Both terms have the same variable part \(a^2\). So, they can be combined.
02

Subtract coefficients

Subtract the coefficients of the like terms. The coefficients are 5 and -6. Subtract these to combine the terms: \(5 - 6 = -1\).
03

Write the simplified expression

Now that we have the coefficient for the combined term, write the simplified expression as \(-1a^2\), or simply \(-a^2\).

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Key Concepts

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

Like Terms
When simplifying algebraic expressions, a critical first step is identifying **like terms**. Like terms are terms that have exactly the same variable component with the same exponents.
For instance, in the expression \(5a^2 - 6a^2\), both terms are "like" because they both contain \(a^2\).
Here are a few examples to help you better understand:
  • \(3x\) and \(5x\) are like terms (both have \(x\)).
  • \(7y^2\) and \(-2y^2\) are like terms (both have \(y^2\)).
  • \(4z\) and \(z\) are like terms (both have \(z\)); even if it鈥檚 written without a coefficient, like \(z\), it's understood to be \(1z\).
Recognizing like terms is essential as it allows you to simplify expressions by combining these terms effectively. You cannot combine terms with different variables or different powers, such as \(2x\) and \(2y\), since they are not like terms.
Coefficients
Understanding **coefficients** is another important part of simplifying algebraic expressions. A coefficient is simply the numerical factor in a term that includes a variable.
For example, in the term \(5a^2\), the coefficient is 5. Meanwhile, in the term \(-6a^2\), the coefficient is -6. Coefficients tell us how many units of a variable term are present.
Here are some examples to further illustrate:
  • In \(8y\), 8 is the coefficient of \(y\).
  • In \(-3x^2\), -3 is the coefficient of \(x^2\).
  • In \(z\), where no coefficient is explicitly written, the coefficient is 1 (since it鈥檚 really \(1z\)).
In algebra, coefficients are crucial for combining like terms, as it is the coefficients of these terms that you add or subtract, not the variable parts. This leads to a simplified expression.
Combining Terms
**Combining terms** is how we simplify algebraic expressions by merging like terms. When you combine like terms, you are essentially performing arithmetic on the coefficients while keeping the variable part the same.
Let's revisit our example: \(5a^2 - 6a^2\). Here, both terms have \(a^2\), making them like terms. To simplify, you subtract the coefficients: \(5 - 6 = -1\), leaving you with \(-1a^2\), or simply \(-a^2\).
Consider these additional examples:
  • \(3x + 7x = (3 + 7)x = 10x\)
  • \(2b^2 - 9b^2 = (2 - 9)b^2 = -7b^2\)
  • \(-4c + c = (-4 + 1)c = -3c\)
Combining terms is about focusing on the numerical coefficients. Maintain the integrity of the variables and powers. This method reduces expressions to their simplest form, making calculations easier and more streamlined.

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