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Chapter 5: Problem 110

Simplify each expression by combining any like terms. $$ -6 z+20-3 z $$

Short Answer

Expert verified
The simplified expression is \(-9z + 20\).

Step by step solution

01

Identify Like Terms

In the expression equation \(-6z + 20 - 3z\), you must first identify the terms that are like terms. Like terms have the same variable raised to the same power. Here, \(-6z\) and \(-3z\) are like terms because they both have the variable \(z\). The term \(20\) is a constant and does not have any like terms in this expression.
02

Combine Like Terms

Combine the like terms by adding their coefficients. The coefficients of \(z\) are \(-6\) and \(-3\), so you add them together: \((-6) + (-3) = -9\). This simplifies the expression to \(-9z + 20\), as there is no change to the constant \(20\).

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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 an essential skill in algebra that involves grouping terms with the same variables. In the expression \[-6z + 20 - 3z\] terms with the variable \(z\) are considered like terms. When identifying like terms, look for terms with the exact same variable raised to the same power, not just any variable. This is crucial because only like terms can be combined. For example,
  • \(-6z\) and \(-3z\) are like terms, as both contain the variable \(z\).
  • The number \(20\) is a constant term, which means it doesn't have a variable, and thus cannot be combined with \(-6z\) or \(-3z\).
When you add these like terms, you add their coefficients (the number in front of a variable) together. This is essential to simplify the expression efficiently and correctly.
Simplifying Expressions
Simplifying expressions is a process of rewriting them in a cleaner or more digestible form. This often involves combining like terms, which reduces the number of terms in an expression and makes it easier to understand or solve. Consider the expression \[-6z + 20 - 3z\] The first step is identifying and combining like terms, as seen with
  • \(-6z + (-3z)\)
By adding the coefficients of like terms, you get \(-9z\) Simplification requires precision. Ensure that only terms that are truly like are combined. With consistent practice, this process becomes second nature. Remember, constants like \(20\) remain unchanged during combining of like terms unless they have their specific like terms present.
Coefficients
Coefficients are numbers placed before and multiplied by variables in algebraic expressions. In an expression like\(-6z + 20 - 3z\), coefficients play a pivotal role in determining the value of each term. For instance,
  • In \(-6z\), \(-6\) is the coefficient of \(z\).
  • Similarly, in \(-3z\), \(-3\) is the coefficient.
When you combine like terms, you are essentially combining their coefficients, which, in this case, are \(-6\) and \(-3\). This addition results in a new coefficient, which is \(-9\),for the term \(-9z\). Understanding coefficients is crucial because they directly affect the outcome of the combined terms and the simplified expression, maintaining the balance and correctness of algebraic manipulation.

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Most popular questions from this chapter

Subtract using a vertical format. $$ \begin{array}{r} 5 x^{3}-4 x^{2}+6 x-2 \\ -\left(3 x^{3}-2 x^{2}-x-4\right) \\ \hline \end{array} $$

Match each expression on the left with its simplification on the right. Not all letters on the right must be used and a letter may be used more than once. a. \(3 y\) b. \(9 y-6 y^{2}\) c. \(10 x\) d. \(25 x^{2}\) e. \(10 x-6\) f. none of these $$ (15 x-3)-(5 x-3) $$

Simplify each expression. Assume that variables represent positive integers. $$ a^{4 m+1} \cdot a^{4} $$

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Which is larger? (See the Concept Check in this section.) a. \(9.7 \times 10^{-2}\) or \(1.3 \times 10^{1}\) b. \(8.6 \times 10^{5}\) or \(4.4 \times 10^{7}\) c. \(6.1 \times 10^{-2}\) or \(5.6 \times 10^{-4}\)
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