Problem 60 - Algebra Form and Function

Question

Write an expression in \(x\) representing the result of the given operations on \(x\). Is the expression linear in \(x ?\) Add \(5,\) multiply by \(2,\) subtract \(x\).

Step-by-Step Solution

Answer

1. Add 5 to x 2. Multiply by 2 3. Subtract x 4. Simplify the expression 5. Determine if the expression is linear in x Answer: Yes, the final expression is linear in x, as it can be written in the form \(ax + b\), where the simplified expression is \(x + 10\).

Step 1: Add 5 to x

First, add 5 to x: \(x + 5\).

Step 2: Multiply by 2

Next, multiply the result from step 1 by 2: \(2(x + 5)\).

Step 3: Subtract x

Finally, subtract x from the result in step 2: \(2(x + 5) - x\).

Step 4: Simplify the expression

Simplify the expression found in step 3: \(2(x + 5) - x = 2x + 10 - x = x + 10\).

Step 5: Determine if the expression is linear in x

An expression is linear in x if it can be represented in the form \(ax + b\). In this case, the simplified expression is \(x + 10\), which can be re-written as \(1x + 10\). Therefore, the expression is linear in x.

Key Concepts

Simplifying Expressions

Simplifying expressions involves combining similar terms and applying basic arithmetic to make an expression neater and easier to read. In this exercise, we start with the operations on the variable, namely adding, multiplying, and subtracting. Here is how it breaks down:

• First, we add 5 to our variable, resulting in the expression \( x + 5 \).
• Then, we multiply this new expression by 2, giving us \( 2(x + 5) \).
• Finally, we subtract \( x \) from what we obtained in the last step, leading to \( 2(x + 5) - x \).

To simplify, distribute 2 into the parentheses: \( 2x + 10 \). After that, subtract \( x \) which results in \( 2x + 10 - x = x + 10 \). Simplification is all about carefully following arithmetic rules and ensuring that you combine like terms.

Operations on Variables

In algebra, carrying out operations on variables forms the foundation for understanding expressions like the one in this exercise. Here’s a breakdown of how these operations are applied:

• **Addition:** Adding 5 to \( x \) is a straightforward operation. It shifts the entire expression up by 5 units on a number line.
• **Multiplication:** Multiplying an expression by 2 stretches it, effectively doubling each component inside the parentheses: \( 2(x + 5) \).
• **Subtraction:** Subtraction introduces a reduction or removal, such as subtracting \( x \) from the doubled expression \( 2x + 10 \), reducing it to \( x + 10 \).

Understanding these operations and how they affect the variable's value helps demystify complex algebraic manipulations. Each operation must adhere to the rules of algebra, ensuring accuracy and coherence in the solution.

Linear Algebra

Linear algebra encompasses concepts where expressions are proportional or linear in their terms. In this context, a linear expression in \( x \) fits the formula \( ax + b \), which is a straight line when graphed on a Cartesian coordinate system.

When we simplify to \( x + 10 \), the expression becomes \( 1x + 10 \). This clearly fits the linear form \( ax + b \), where \( a = 1 \) and \( b = 10 \). Such expressions indicate a direct proportion: as \( x \) increases, the entire expression increases in a predictable, linear fashion.

In linear algebra, understanding these forms helps in graphing, solving equations, and applying mathematical models. The simplicity of linear terms makes them foundational in algebra and higher mathematics, as they describe constant rates of change.