Problem 48 - Prealgebra

Question

Simplify each expression. $$6 z-3-10 z+7$$

Step-by-Step Solution

Answer

The simplified expression is \(-4z + 4\).

Step 1: Combine Like Terms for 'z'

First, identify the terms that include the variable \( z \). In this expression, these terms are \( 6z \) and \( -10z \). Combine these terms by performing the addition: \( 6z - 10z \). This results in \( -4z \).

Step 2: Combine Constant Terms

Next, find the constant terms (terms without variables). In this expression, the constant terms are \( -3 \) and \( +7 \). Combine these: \( -3 + 7 \), which simplifies to \( 4 \).

Step 3: Write the Simplified Expression

Now that you've combined both variable and constant terms, write down the simplified expression. From Steps 1 and 2, we have \( -4z \) and \( 4 \). Combine these to form the final simplified expression: \( -4z + 4 \).

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, a crucial step involves combining like terms. Like terms are terms that have the same variables raised to the same power. For instance, in the expression \(6z - 3 - 10z + 7\), the like terms involving the variable \(z\) are \(6z\) and \(-10z\). Like terms can be combined because they represent the same types of quantities. To combine them, you simply add or subtract their coefficients (the numerical part of the term). For example, \(6z - 10z = -4z\) because you subtract the coefficient of \(-10z\) from the coefficient of \(6z\), which results in \(-4z\). This process helps in simplifying expressions and makes it easier to understand and solve problems. Additionally, combining constant terms (terms without variables) is necessary. In this problem, \(-3\) and \(+7\) are constants that can be combined to yield a sum of \(4\). Engaging in this combining process is essential in prealgebra to prepare for more complex algebraic manipulations later on.

Polynomials

A polynomial is a mathematical expression consisting of variables, coefficients, and constants, combined using operations of addition, subtraction, and multiplication. In simple terms, a polynomial is a sum of many terms. The expression \(6z - 3 - 10z + 7\) is a polynomial because it can be written as a combination of terms involving the variable \(z\) and constants. Key aspects of polynomials include the degree, which is the highest power of the variable in the expression. Understanding polynomials is crucial as it lays the groundwork for more advanced algebra topics. A polynomial like the one in our exercise can be simplified by combining like terms, as demonstrated in the previous section. After simplifying, it results in a shorter, more manageable expression: \(-4z + 4\). This simplified form gives clearer insight into the expression's behavior and aids in solving algebraic equations.

Prealgebra

Prealgebra serves as a foundational building block for all of algebra and higher mathematics. It introduces students to basic concepts that are crucial for understanding more advanced topics later on. Prealgebra focuses on arithmetic operations, introducing variables, and basic problem-solving skills. In prealgebra, students learn to manipulate expressions like \(6z - 3 - 10z + 7\) by understanding the language of mathematics. They become familiar with variables, coefficients, constants, and the importance of expression simplification. This stage of mathematics education organizes how students think about numbers and variables logically, paving the way for the study of algebra. Prealgebra empowers students with the skills to simplify expressions, solve equations, and understand relationships between numbers—a critical step for future success in mathematics.