Problem 32 - Elementary Algebra

Question

Simplify the opposite of the expression \(6 x+5 x-1\).

Step-by-Step Solution

Answer

The opposite of the expression is \(-11x + 1\).

Step 1: Understand the Problem

We need to simplify the opposite of the expression \(6x + 5x - 1\). Simplifying an expression means to combine like terms, and finding the opposite means multiplying the entire expression by -1.

Step 2: Combine Like Terms

First, we combine the like terms in the expression \(6x + 5x - 1\). The terms \(6x\) and \(5x\) are like terms. Add them together: \(6x + 5x = 11x\). Now the expression is simplified to \(11x - 1\).

Step 3: Find the Opposite

To find the opposite of the expression \(11x - 1\), multiply the entire expression by -1: \(-(11x - 1) = -11x + 1\).

Step 4: Final Expression

After multiplying by -1, we simplify it to \(-11x + 1\). This is the opposite of the original expression \(6x + 5x - 1\).

Key Concepts

Combining Like Terms

In algebra, combining like terms is an essential step in simplifying expressions. It involves adding or subtracting terms that have the same variable to a single term. In our exercise, the expression is \(6x + 5x - 1\). Here, \(6x\) and \(5x\) are like terms because they both have the variable \(x\).

When combining these terms, you add their coefficients, the numbers in front of the variables:
- Add 6 and 5 to get \(11x\).
So the expression simplifies to \(11x - 1\).

This step makes it easier to manage and further simplify the expression. Always ensure that you only combine terms with the exact same variable and exponent.

Opposite of Expression

Finding the opposite of an expression means you are performing the operation that gives you the additive inverse. The inverse essentially cancels out the original when added to it, resulting in zero. To do this, multiply the entire expression by -1.

For example, consider the expression \(11x - 1\). To find its opposite, multiply each term by -1:
- \(11x\) becomes \(-11x\)
- \(-1\) becomes \(+1\)

The opposite of the expression \(11x - 1\) is therefore \(-11x + 1\). This operation is vital when solving equations, as it simplifies the manipulation and solution processes of algebraic expressions.

Algebraic Expressions

Algebraic expressions are formulas composed of numbers, variables, and operations (like addition and subtraction). They are used to represent mathematical concepts in a generalized way.

For instance, the expression \(6x + 5x - 1\) contains:
- Coefficients, like 6 and 5, which are numbers in front of the variables.
- Variables, such as \(x\), representing unknown quantities.
- Constants, like \(-1\), which are fixed numbers.

Understanding each component helps in performing operations like simplifying or finding opposites. Algebraic expressions can be manipulated in various ways to solve equations or model real-world situations.