Problem 62 - 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 \(x,\) multiply by \(5,\) subtract \(2 .\)

Step-by-Step Solution

Answer

Add x to x, multiply the result by 5, and then subtract 2. Answer: Yes, the expression is linear in x.

Step 1: Add x

Add x to the existing x value: Expression: \(x + x\)

Step 2: Multiply by 5

Multiply the result from Step 1 by 5: Expression: \((x + x) \times 5\)

Step 3: Subtract 2

Subtract 2 from the result of Step 2: Expression: \((x + x) \times 5 - 2\)

Step 4: Simplify the expression

Combine like terms and simplify the expression: Expression: \(2x \times 5 - 2 \Rightarrow 10x - 2\)

Step 5: Determine if the expression is linear

Our expression is \(10x - 2\), which follows the pattern ax+b, with a=10 and b=-2. Therefore, the expression is linear in x.

Key Concepts

Algebraic Operations

Algebraic operations are steps where you perform mathematical actions on expressions, such as addition, subtraction, multiplication, or division. They help us manipulate expressions to achieve a particular form or solve equations.
When dealing with variables such as \(x\), operations become essential in expressing complex relationships. In our example, we began by adding \(x\) to itself: \(x + x\). This operation combines like terms and is a foundation for understanding variable manipulation.

• **Addition** combines values or variables.
• **Multiplication** amplifies the result of an expression, as seen when \((x + x)\) is multiplied by 5.
• **Subtraction** reduces the expression, as observed when 2 is subtracted in the final step.

Understanding these operations allows you to transform and explore expressions effectively.

Expression Simplification

Simplifying expressions is a crucial step, especially when dealing with multiple operations. The goal is to express the mathematical statement in its simplest form, making it easier to interpret and work with.
The original expression, \((x + x) \times 5 - 2\), was simplified by first combining terms: \(x + x = 2x\). After this combination, multiplication and subtraction were performed in the expression \(2x \times 5 - 2\).

• **Combine like terms** first, which means summing variables or constants that have the same characteristics.
• **Perform operations** according to the order of operations: parentheses, exponents, multiplication/division (left to right), and addition/subtraction (left to right).

The final simplification step results in \(10x - 2\), a more concise version of the initial problem.

Linear Equations

Linear equations represent mathematical statements that express equalities through linear expressions. These equations always graph as straight lines when plotted and take the general form \(ax + b = c\).
In the expression \(10x - 2\), we see that it fits the linear form \(ax + b\) where \(a = 10\) and \(b = -2\). This tells us it is a linear expression. Linear equations are characterized by:

• **Constant rate of change** – as \(x\) changes, the value of the expression increases or decreases at a constant rate due to the coefficient \(a\).
• **No powers or roots of x** – these would make the expression non-linear.
• **Graph as straight lines** – confirming linearity when plotted.

Recognizing whether an expression or an equation is linear helps in predicting behavior and finding solutions.