Problem 92 - College Algebra

Question

Write three equations that are equivalent to \(x=5\)

Step-by-Step Solution

Answer

The three equivalent equations to \(x = 5\) can be represented as \(2x - 5 = 5\), \(10x - 5x = 5x\) and \((x+3) - 3 = 5\).

Step 1: Identify the original equation

The original equation that we'll need to create equivalents for is \(x = 5\). All equivalent equations must resolve to this solution.

Step 2: Create equivalent equations

To construct equivalent equations, one can add, subtract, multiply or divide (except by zero) the same number on both sides of the equation. Below are the three equivalent equations: Equation 1: \(2x - 5 = 5\), Equation 2: \(10x - 5x = 5x\), Equation 3: \((x+3) - 3 = 5\).

Step 3: Validate the equivalent equations

To ensure that the results are accurate, solve each equation. The result should be \(x = 5\). Equation 1: \(2x - 5 = 5 => 2x = 10 => x = 5\). Equation 2: \(10x - 5x = 5x => 5x = 5x => x = 5\). Equation 3: \((x + 3) - 3 = 5 => x = 5\). It's confirmed all equations are equivalent to each other.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra that allows you to find the value of unknown variables. It involves manipulating the equation until you isolate the variable on one side. In this exercise, the original equation was \(x = 5\). To solve similar equations and ensure they're equivalent, you can perform operations that do not change the value of the variable, such as:

• Adding or subtracting the same number from both sides
• Multiplying or dividing both sides by the same non-zero number

These operations maintain the balance of the equation. For instance, if you add 3 to both sides, you can later subtract 3 to return to the original state.

Algebraic Manipulation

Algebraic manipulation involves using arithmetic operations to rearrange and simplify equations. The goal is to create equivalent forms of equations. In the given example, algebraic manipulation was used to construct three equivalent equations:

• First, by handling and arranging terms, like in \(2x - 5 = 5\)
• Second, simplifying terms, as seen in \(10x - 5x = 5x\)
• Finally, transposing and balancing equations, demonstrated in \((x+3) - 3 = 5\)

These steps show how manipulating terms strategically results in equations that appear different but are truly equivalent.

Identity Property of Equations

The identity property of equations states that something remains the same when equal transformations are applied. This property is crucial for validating that two or more equations yield the same solution. In the case of equivalent equations like \(x = 5\), adopting this property helps verify consistency across various forms:

• Each equation should resolve back to \(x=5\)
• Ensures that any manipulative steps have not altered the core solution
• Reinforces the equality of all forms actually equaling the variable's intended value

This understanding helps simplify and verify complex algebraic operations while maintaining equation integrity.