Problem 12 - Elementary Algebra

Question

Evaluate. \(2 x-3,\) where \(x=-3\)

Step-by-Step Solution

Answer

The expression evaluates to -9 when \(x = -3\).

Step 1: Understand the Expression

We have the expression \(2x - 3\). Our task is to evaluate this expression when \(x = -3\). This means we'll substitute \(-3\) for \(x\) in the expression.

Step 2: Substitute the Variable

Replace \(x\) in the expression with \(-3\). This changes the expression from \(2x - 3\) to \(2(-3) - 3\).

Step 3: Perform the Multiplication

Calculate the result of the multiplication part of the expression. So, multiply \(2\) by \(-3\), which equals \(-6\). Thus, the expression becomes \(-6 - 3\).

Step 4: Perform the Subtraction

Subtract \(3\) from \(-6\). The result of \(-6 - 3\) is \(-9\).

Key Concepts

Substitution in Algebra

Substitution is a vital concept in algebra that makes solving expressions and equations easier. It involves replacing a variable in an expression with a specific value. This is done to simplify the expression or to calculate a precise numerical result. For instance, in the expression \(2x - 3\), instead of keeping \(x\) as a variable, we substitute it with the given number, like \(-3\) in our example.Doing this involves simply replacing every instance of the variable with the number. After the replacement, the expression becomes entirely numerical, making it easier to compute. This technique is broadly used in various types of algebraic problems, serving as a foundational skill for more complex algebraic manipulations.

Algebraic Expressions

An algebraic expression consists of numbers, variables, and arithmetic operations, like addition, subtraction, multiplication, and division. In the expression \(2x - 3\), \(2x\) represents a term where the number 2 (coefficient) is multiplied by the variable \(x\). The expression is simplified by performing operations according to algebraic rules.

• Terms are separated by plus or minus signs.
• The coefficient is the numerical part attached to a variable.
• Constant terms are numbers without variables, like \(-3\) here.

Algebraic expressions are cornerstones of algebra and are critical for forming equations, inequalities, and more complex mathematical constructs.

Basic Arithmetic Operations

These operations include addition, subtraction, multiplication, and division, which are fundamental to evaluating expressions. In our specific example, evaluating \(2x - 3\) involves using multiplication and subtraction.First, multiplication: once we substitute \(x = -3\) into the expression, we perform the multiplication \(2 \times (-3)\), resulting in \(-6\). Next is subtraction, where \(-6 - 3\) equals \(-9\). Understanding these basic operations helps in breaking down complex expressions into manageable pieces.

• Multiplication often precedes addition or subtraction in the order of operations.
• Subtraction is taking away or comparing values to find the difference.

Mastering basic arithmetic is crucial for exploring more advanced mathematical concepts and solving algebraic expressions efficiently.