Problem 63 - College Algebra Essentials

Question

Solve equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. \(3(x+2)=7+3 x\)

Step-by-Step Solution

Answer

The solved equation is \(6 = 7\), which is false and hence an inconsistent equation.

Step 1: Distributive Property

First, use the distributive property on the left-hand side of the equation: \(3 \cdot x + 3 \cdot 2 = 7 + 3x\). This simplifies to \(3x + 6 = 7 + 3x\).

Step 2: Isolating Variables

Next, isolate all variable terms on one side of the equation and constant terms on the other. Here, after subtraction of \(3x\) from both sides the equation remains the same: \(3x + 6 - 3x = 7 + 3x - 3x\), simplifying to \(6 = 7\).

Step 3: Identify the type of equation

Lastly, by looking at the final equation, \(6 = 7\), it becomes clear that it isn't true, hence it's an inconsistent equation because inconsistent equations have no solutions as these are false statements.

Key Concepts

The Distributive Property

The distributive property is a fundamental concept in algebra that helps streamline expressions and equations. This property takes a factor multiplied by a sum or difference and distributes the multiplication across each term inside the parentheses.
For the equation given in the exercise, the left-hand side starts as \(3(x + 2)\). We apply the distributive property here to break it down into two separate products: \(3 \cdot x + 3 \cdot 2\). This simplifies into \(3x + 6\).

• The distributive property is represented by \(a(b + c) = ab + ac\).
• It helps in removing parentheses to simplify an equation.

By using this property, algebraic equations are often easier to manipulate, making future steps like isolating variables more straightforward.

Understanding Inconsistent Equations

An inconsistent equation is one that leads to a false statement, meaning there is no solution that satisfies the equation. When you solve an equation and end up with something nonsensical, such as \(6 = 7\), it’s identified as inconsistent. These equations are crucial in understanding the types of solutions equations can have.
In the original exercise, after applying the distributive property, we attempted to isolate the variables. The simplified form of the equation resulted in \(6 = 7\), clearly illustrating an inconsistency.

• An inconsistent equation is significant since it implies that there aren’t any values for the variables that can satisfy it.
• Recognizing an inconsistent equation helps avoid further manipulation, saving time and effort.

Knowing when an equation is inconsistent is essential for correctly determining its nature.

Algebraic Equations Simplified

Algebraic equations are mathematical statements indicating equality between two expressions containing variables and constants. The main goal in solving these is to find the value of the unknowns that make the equation true. There are a few types of solutions these equations might have:

• Conditional Equations: Have specific solutions where certain values satisfy the equation.
• Identities: True for all values of the variables involved.
• Inconsistent Equations: As discussed previously, these have no solutions.

Algebra frequently requires recognizing these forms and understanding their characteristics.
In this exercise, we explored whether the given equation was a conditional equation, an identity, or an inconsistent equation, concluding it was inconsistent.