Problem 63 - Algebra and Trigonometry

Question

Solve each 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 equation \(3(x+2)=7+3x\) is an inconsistent equation, as no value of \(x\) can make the equation true.

Step 1: Simplify the equation

Start by simplifying both sides of the equation. In this case, distribute \(3\) in the expression \(3 * (x + 2)\) on the left side: \[3x + 6 = 7 + 3x\]

Step 2: Solve for x

Next, let's try to solve for \(x\) by subtracting \(3x\) from both sides to get the variable on one side and constants on the other: \[6 = 7\]

Step 3: Evaluate the equation

We notice that the equation evaluates to an incorrect statement where \(6\) is equal to \(7\), which is not true. This means that there are no values of \(x\) that satisfy the equation. Hence the equation is inconsistent.

Key Concepts

Understanding Algebraic Equations

Algebraic equations are mathematical statements that assert equality between two expressions. These expressions typically involve variables, constants, and arithmetic operations. For example, the equation \(3(x+2)=7+3x\) is an algebraic equation because it includes the variable \(x\), a constant \(7\), and an operation, namely multiplication and addition. All these elements are combined to describe a relationship.

In algebra, solving these types of equations means finding the values of the variables that make the equation true. This process involves simplifying the expressions, isolating the variable on one side, and performing operations to find the variable's value. However, not all algebraic equations will have a solution, leading us to explore concepts like conditional and inconsistent equations.

Conditional Equations Explained

A conditional equation is an equation that is true only for specific values of the variable involved. This means that only certain numbers, when substituted into the equation for the variable, will satisfy the equation.

• The key feature of a conditional equation is that it has one or more specific solutions.
• If simplifying an equation leads to an expression like \(x = 5\), our equation is conditional because it tells us exactly what \(x\) must be.

For example, solving \(2x + 3 = 9\) leads to \(x = 3\) because when \(x\) is replaced with \(3\), both sides of the equation equal \(9\). Thus, this specific value satisfies the condition of the equation. Understanding this is crucial in equation solving as it requires logic to isolate variables and find these solutions.

Inconsistent Equations: No Solution Possible

An inconsistent equation is one that has no solution because the simplified form results in a false statement. This happens when the process of solving leads to a contradiction — meaning the variable terms cancel out and leave behind an impossible equality.

• For example, in our initial equation, \(3(x+2)=7+3x\), the simplifying steps lead to \(6 = 7\).
• Since \(6\) does not equal \(7\), this incorrect statement shows the equation is inconsistent.

This means no value of the variable will make the equation true. Encountering such equations during algebra practice indicates a deeper understanding as recognizing these situations can save time by not searching for non-existent solutions.