Problem 111 - Algebra A Combined Function

Question

Simplify each expression by combining any like terms. $$ 7 x+2-8 x-6 $$

Step-by-Step Solution

Answer

The simplified expression is \(-x - 4\).

Step 1: Identify Like Terms

Look at the expression: \( 7x + 2 - 8x - 6 \). Identify the like terms. Like terms are those that have the same variable or are constants. In this case, \(7x\) and \(-8x\) are like terms because they both have the variable \(x\). Similarly, \(2\) and \(-6\) are constant terms and can be combined.

Step 2: Combine the Like Terms with Variables

Focus on the variable terms \(7x\) and \(-8x\). Combine them by adding their coefficients: \(7 - 8 = -1\). So, \(7x - 8x = -1x\) or simply \(-x\).

Step 3: Combine the Constant Terms

Now, look at the constants \(2\) and \(-6\). Combine them by adding: \(2 - 6 = -4\).

Step 4: Write the Simplified Expression

Combine the results from Steps 2 and 3 to write the simplified expression. The expression becomes \(-x - 4\).

Key Concepts

Like Terms

In algebra, like terms are terms that have the same variable raised to the same power. They can be combined easily because they share common characteristics. For example, in the expression \( 7x + 2 - 8x - 6 \), the terms \( 7x \) and \( -8x \) are like terms since they both contain the variable \( x \). Similarly, the numbers \( 2 \) and \( -6 \) are also considered like terms because they don't contain any variables, making them constants. Recognizing like terms is crucial when simplifying expressions. It helps you group and manage terms efficiently without altering the mathematical meaning of the expression.

Combining Terms

Once you have identified like terms in an expression, the next step is to combine them. Combining terms involves adding or subtracting the coefficients of like terms. Coefficients are the numerical parts attached to variables in a term. For instance, in the terms \( 7x \) and \( -8x \), the coefficients are \( 7 \) and \( -8 \) respectively. Combining these terms means performing the arithmetic operation: \( 7 - 8 \), which equals \( -1 \). Therefore, \( 7x - 8x \) simplifies to \( -x \) because \( -1x \) is typically written as \( -x \). Combining terms enables you to produce a cleaner and simpler expression.

Constants

Constants are numbers that stand alone without attached variables in an algebraic expression. They remain the same as they don't change in value. In the expression \( 7x + 2 - 8x - 6 \), the constants are \( 2 \) and \( -6 \).When simplifying expressions, it's essential to also combine constants by performing straightforward arithmetic operations. Here, you would add the constants: \( 2 - 6 \), which simplifies to \( -4 \). Combining constants gives you one simplified number, which is then added to the simplified expressions of any variable terms you have.

Coefficients

Coefficients are key components in algebraic expressions as they signify how many times a term should be counted. They are the numbers located before a variable, such as \( 7 \) in \( 7x \) or \( -8 \) in \( -8x \). They tell you how much of each variable you have.When simplifying expressions, you'll primarily work with coefficients as you combine like terms. If an expression features \( 7x \) and \( -8x \), it means you have \( 7 \) of \( x \) minus \( 8 \) of \( x \), ultimately resulting in \( -1 \) of \( x \), or \( -x \). Understanding and managing coefficients help streamline and simplify algebraic calculations effectively.