Problem 41 - College Algebra with Modeling and Visualization

Question

Complete the following. (a) Solve the equation symbolically. (b) Classify the equation as a contradiction, an identity, or a conditional equation. $$ 3(x-1)=5 $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{8}{3} \). The equation is conditional.

Step 1: Distribute 3 in the Equation

Start by distributing the 3 to both terms inside the parenthesis. The equation becomes: \[ 3x - 3 = 5 \].

Step 2: Isolate the Variable Term

Add 3 to both sides of the equation to move the constant term to the right-hand side. \[ 3x - 3 + 3 = 5 + 3 \] This simplifies to: \[ 3x = 8 \].

Step 3: Solve for x

Divide both sides by 3 to solve for \( x \): \[ x = \frac{8}{3} \].

Step 4: Determine Type of Equation

Since the equation has a single solution \( x = \frac{8}{3} \), it is a conditional equation. Conditional equations are only true for specific values of the variable.

Key Concepts

Solving Equations

Solving equations is all about finding the value of the variable that makes the equation true. An equation is made up of two expressions connected by an equal sign, indicating that the values on both sides are equal. To solve an equation, you aim to manipulate it such that the variable (\(x\)) is isolated on one side.

In the exercise given, you'll notice we start with \(3(x-1)=5\). Our goal is to find what \(x\) should be, so that both sides have equal value
. Common steps in solving equations include:

• Distributing any constants or coefficients
• Collecting like terms
• Adding or subtracting numbers to both sides to maintain balance
• Dividing or multiplying to finally solve for the variable

Using these steps ensures that the operations you perform are reversible, keeping the equation valid until the solution is reached.

Distributive Property

The distributive property is a key concept used in algebra to simplify expressions and equations. It involves distributing a multiplication operation over addition or subtraction within parentheses. The property states that \(a(b+c) = ab + ac\).

In our example, we utilize this property to write out the expression in \(3(x-1)=5\) as \(3x - 3 = 5\). Here, the number outside the parentheses (3) is multiplied by each term inside (both \(x\) and \(-1\)). This helps simplify the equation, making it easier to isolate the variable.

Remember, the distributive property is crucial for breaking down complex expressions and is often your first step if you encounter parentheses in an equation.

Types of Equations

Equations can be categorized based on their solutions. Understanding these categories helps in identifying what kind of solution you're dealing with. There are generally three types of equations:

• **Contradictions**: These have no solution. The equation results in a false statement, like \(2=3\).
• **Identities**: These hold true for all values of the variable, such as \(x=x\) or \(3x=3x\).
• **Conditional Equations**: These are true only for specific values of the variable.

For the exercise provided, the equation \(3(x-1)=5\) has one specific solution \(x = \frac{8}{3}\), classifying it as a conditional equation. Conditional equations are the most common, often involving variables that need solving to find the particular value that satisfies the equation.