Problem 110 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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 \(20\).

Key Concepts

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.