Problem 86 - Introductory Algebra for College Students

Question

Simplify: \(x-0.3 x\).

Step-by-Step Solution

Answer

The simplified form of the given expression is \(0.7x\).

Step 1: Combine Similar Terms

Given the equation \(x - 0.3x\), note that there are two terms here that are alike: \(x\) and \(-0.3x\). Both are basically 'x' multiplied by some value. Therefore, they can be combined by subtraction using the rule of combining like terms in an equation. Subtract 0.3x from x.

Step 2: Identify the operation required

Determine what computation or simplification is needed.

Step 3: Perform the calculation

Execute the computation step by step.

Step 4: Simplify and state the result

Simplify and write the final answer.

Step 5: Conclude with the answer

The simplified form of the given expression is \(0.7x\).

Key Concepts

Combining Like Terms

In algebra, combining like terms is a fundamental concept used to simplify expressions and equations. Like terms are terms in an expression that have the same variable raised to the same power. This concept allows us to simplify complex expressions and make calculations more straightforward.

When we look at the original exercise, the expression given is \(x - 0.3x\). Here, both terms involve the variable \(x\). This means they are like terms because both have the same variable part, with no other factors or powers involved. To combine like terms:

• Look for terms that have the exact same variable and exponent.
• Add or subtract the coefficients of these terms.
• Keep the variable and the exponent unchanged in the combined term.

In our example, the coefficients are 1 (implied in \(x\)) and -0.3 for \(0.3x\). So, we subtract the coefficients: \(1 - 0.3 = 0.7\). This results in the simplified term \(0.7x\). By combining like terms, the expression becomes simpler, helping us to continue solving or working with the equation.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves performing operations that reduce them to simpler, more manageable forms. This process helps us see the underlying structure of the expression and may reveal solutions more clearly.

Let's break down the steps:

• First, remove any unnecessary parentheses by distributing multiplication over addition or subtraction if needed.
• Then, perform any arithmetic operations possible, such as combining like terms or simplifying fractions.
• Finally, arrange the expression in a standard form, often by writing terms in decreasing power order or by factoring if possible.

In the exercise \(x - 0.3x\), simplifying involved combining the terms, as explained previously. Once combined, the simplified expression was \(0.7x\). This forms a cleaner, more workable equation, allowing for further calculations or solutions. Simplified expressions are easier to understand and manipulate, which is why this step is essential in algebra.

Linear Equations

Linear equations are equations of the first degree, which means they involve variables raised to the power of one. These equations are foundational in algebra because they describe a straight-line relationship between variables.

A standard form for a linear equation in one variable is \(ax + b = 0\), where \(a\) and \(b\) are constants. Solving these equations is straightforward and involves isolating the variable on one side of the equation.

When simplifying algebraic expressions like \(x - 0.3x\), you might be working towards creating or solving a linear equation. It involves reducing complexity so that relationships between variables become apparent. In our exercise, although it's not a full equation, simplifying \(x - 0.3x\) into \(0.7x\) is a step toward potentially using it in a larger context, such as an equation where solving for \(x\) is necessary.

Linear equations are used across various fields to model real-world phenomena, making understanding and simplifying such expressions invaluable.