Problem 75 - Algebra 2

Question

Simplify each expression. $$ 2 x+9 y+4 z-y-8 x $$

Step-by-Step Solution

Answer

The simplified expression is \(-6x + 8y + 4z\).

Step 1: Identify Like Terms

To simplify the expression, begin by identifying terms that can be combined. The expression is \(2x + 9y + 4z - y - 8x\). Here, \(2x\) and \(-8x\) are like terms, and \(9y\) and \(-y\) are like terms.

Step 2: Combine Like Terms for 'x'

Combine the like terms for \(x\): \(2x - 8x\). This simplifies to \(-6x\).

Step 3: Combine Like Terms for 'y'

Combine the like terms for \(y\): \(9y - y\). This simplifies to \(8y\).

Step 4: Combine All Terms

Put all simplified terms together. The expression becomes \(-6x + 8y + 4z\).

Key Concepts

Like Terms

In algebra, one of the key concepts is understanding what "like terms" are. This is crucial when simplifying algebraic expressions. Like terms can be defined as terms that contain the same variable raised to the same power. For example, in the expression \(2x + 9y + 4z - y - 8x\), the terms \(2x\) and \(-8x\) are like terms because they both involve the variable \(x\). Similarly, \(9y\) and \(-y\) are like terms since they involve the variable \(y\).

It is important to note that the coefficients (the numerical part of the terms) may be different, but as long as the variables and their exponents are identical, they are considered like terms. This understanding is the foundation of combining terms, which is a critical step in simplifying algebraic expressions.

Combining Terms

After identifying like terms in an expression, the next step is combining these terms. This process essentially means performing the arithmetic operation indicated (usually addition or subtraction) on the coefficients of the like terms. Let's look at our example: the expression \(2x + 9y + 4z - y - 8x\).

First, you combine the like terms \(2x\) and \(-8x\). Subtract \(8x\) from \(2x\) which results in \(-6x\). Next, take the like terms \(9y\) and \(-y\), and combine them by subtracting \(y\) from \(9y\) to get \(8y\).

This step helps in condensing the expression, making it simpler and more manageable. By combining terms, you reduce the number of terms in the expression while keeping its original value, which is crucial in solving algebraic problems efficiently.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operators (such as addition and subtraction) that represent values. For example, \(2x + 9y + 4z - y - 8x\) is an algebraic expression. These expressions do not always represent single values, as they can vary depending on the values assigned to the variables.

Algebraic expressions are the foundation of algebra. Simplifying them involves reducing the expression to its most concise form while retaining its equivalence. This makes it easier to understand and solve problems involving expressions.

• They contain variables (like \(x\), \(y\), \(z\)) that can stand in for unknown values.
• They use numbers as coefficients, which multiply the variables.
• The operators within the expressions indicate the operations to perform among the terms.

Understanding algebraic expressions and how to manipulate them is crucial for solving equations and inequalities.