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Chapter 3: Problem 77

Simplify by combining like terms. $$-10 t^{2}+9 t^{2}$$

Short Answer

Expert verified
Question: Simplify the expression $$-10 t^{2} + 9 t^{2}$$. Answer: $$-t^{2}$$.

Step by step solution

01

Identify like terms

We are given the expression $$-10 t^{2} + 9 t^{2}$$. Since both terms have the same power of variable t, they are like terms.
02

Combine like terms

To simplify, add the coefficients of the like terms, which are -10 and 9. So we have $$(-10 + 9) t^{2}$$.
03

Simplify the expression

Now, simplify the expression by doing the addition operation in the brackets: $$(-10 + 9) t^{2} = (-1) t^{2}$$. So the simplified expression is $$-t^{2}$$.

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Key Concepts

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

Simplifying Algebraic Expressions
Simplifying algebraic expressions is all about making them shorter and easier to work with, without changing their value. It involves reducing the complexity by combining like terms and performing operations. In algebra, expressions can often look long and confusing, but the goal when simplifying is to break them down into their simplest form. This not only helps in understanding the problem better but also in solving it faster.

To simplify an expression, follow these steps:
  • Identify and collect like terms (which we'll discuss next)
  • Combine the coefficients of the like terms
  • Perform any remaining operations such as addition or subtraction
By doing so, expressions become more manageable and easier to interpret. In essence, simplifying doesn't change the underlying meaning of the equation; it just makes it clearer.
Identifying Like Terms
Like terms in algebra are terms that have the same variables raised to the same power. The only difference between like terms is their coefficients. Recognizing like terms is crucial for combining and simplifying expressions. For example, in the expression \(-10 t^{2} + 9 t^{2}\), both terms contain \(t^{2}\). Therefore, they are like terms.

Here are some tips for identifying like terms:
  • Look for terms with the exact same variables and powers
  • Constants (numbers without variables) are also like terms with each other
  • Be cautious with signs; they affect the coefficients
Once you've spotted the like terms in an expression, you can easily proceed to add or subtract their coefficients, an essential step in simplifying the expression.
Coefficients in Algebra
Coefficients are the numerical part of terms in an expression. They tell us how many times the variable should be multiplied. Understanding coefficients is key to combining terms. In the expression \(-10 t^{2} + 9 t^{2}\), the coefficients are \(-10\) and \(9\).

To work effectively with coefficients:
  • Recognize that they multiply the variable they are attached to
  • Only combine coefficients of like terms
  • Remember that coefficients can be positive or negative, influencing the result of operations
When you combine coefficients of like terms by adding or subtracting, you're effectively merging them into a single term. For instance, for the expression \(-10 t^{2} + 9 t^{2}\), combining these gives \((-10 + 9)t^{2}\), simplifying to \(-t^{2}\). Coefficients guide the simplification process, enabling the transformation of complex expressions into simpler forms.

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