Problem 2 - College Algebra and Calculus: An Applied Approach

Question

Determine whether the equation is an identity or a conditional equation. $$ 3(x+2)=3 x+6 $$

Step-by-Step Solution

Answer

The equation '3(x + 2) = 3x + 6' is an identity because it is true for all values of 'x'.

Step 1 Distribute

Start by distributing '3' on the left side of the equation to get: '3*x + 3*2 = 3x + 6' which simplifies to '3x + 6 = 3x + 6'.

Step 2 Compare both sides

Now, compare both sides of the equation. If they exactly match with each other, this means that the equation will be valid for any value of 'x', and hence, it will be an identity. In this case, '3x+6' exactly matches with '3x+6' on the right hand side.

Key Concepts

Conditional Equations

Understanding whether an equation is a "conditional equation" helps you categorize it by its solutions. Unlike identity equations, which hold true for any value of the variable, conditional equations are only valid or true for specific values of the variable. When solving equations, it’s essential to determine if the solution fits in every case (identity) or just certain cases (conditional). To identify a conditional equation, isolate the variable and solve the equation using algebraic techniques. This will often involve:

• Performing operations such as addition, subtraction, multiplication, or division.
• Using inverse operations to get the variable by itself on one side of the equation.

When you solve a conditional equation, the result provides you with specific values (or sometimes a value) that satisfy the equation. If the equation was to balance perfectly for all values of 'x', then you would have an identity rather than a conditional equation. In our example, both sides equal each other for any 'x', meaning it is not conditional but rather an identity.

Distributive Property

The distributive property is a fundamental rule in algebra that simplifies expressions and solves equations efficiently.In the equation given as an example, we see this property at work: 3(x+2) becomes 3*x + 3*2. This property allows you to multiply a single term by each term inside a bracket (or parentheses). It is expressed generally as:\(a(b + c) = ab + ac\)This rule is crucial because it helps to simplify complex expressions and prepare them for further manipulation or solving. Without applying the distributive property, an equation may appear more complicated than it actually is.Using the distributive property:

• Assists in solving multi-step equations by breaking them into simpler parts.
• Aids in simplifying expressions to see if they match on both sides of an equation.

Thus, understanding this property is key when analyzing the structure of algebraic equations like the one in the exercise.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that stand in for a specific value. They can include:

• Constants, which are numbers on their own.
• Variables, which are symbols like 'x' that represent unknown values.
• Operators, which are mathematical signs like '+', '-', '*', and '/'.

In our example, both sides of the equation '3(x+2)=3x+6' are algebraic expressions. When you compute these expressions, you use the rules and properties of algebra to find equivalence or solutions. Understanding algebraic expressions involves:

• Recognizing how expressions can be manipulated using variables.
• Appreciating that expressions can often be rewritten or simplified to illustrate equivalence.
• Simplifying equations and checking for possible identities or solutions.

Algebraic expressions and their manipulation are foundational to mathematics, enabling you to solve problems and verify solutions, just like confirming identity or conditional equations.