Problem 78 - College Algebra

Question

What are equivalent equations? Give an example.

Step-by-Step Solution

Answer

Equivalent equations are equations that, though they may look different, hold the same solutions. For instance, the equations \(2x - 6 = 4\) and \(2x = 10\) are equivalent because they both have the same solution, \(x = 5\).

Step 1: Understanding Equivalent Equations

Equivalent equations are essentially different algebraic equations that hold the same solutions. In other words, whatever values satisfy one equation will satisfy the other as well.

Step 2: Example of Equivalent Equations

For example, assume we have the equation \(2x - 6 = 4\). If we add 6 to both sides, we get a new equation, \(2x = 10\). These two equations are equivalent because they have the same solutions. We can confirm this by solving for \(x\). For the initial equation, adding 6 to both sides gives \(2x = 10\), dividing by 2 gives \(x = 5\). For the second equation, just divide by 2 and get the same solution, \(x = 5\). So any value that satisfies the first equation (here, \(5\)) also satisfies the second equation.

Key Concepts

Algebraic Equations

At the heart of algebra, algebraic equations are the tools that allow us to solve problems involving unknown quantities. An algebraic equation is a statement of equality between two expressions that contain variables, constants, and operational symbols. For instance, the equation
\( 2x + 3 = 7 \) contains the variable \(x\), constants 2, 3, and 7, and the operation of addition.

To comprehend the importance of these equations, imagine a scales balance: the two pans of the balance represent the two sides of the equation. The goal in solving an algebraic equation is to find the value(s) of the variable(s) that make the equation 'balance', meaning the expressions on both sides of the equality sign have the same value. This foundational concept leads us to the art of solving these algebraic mysteries.

Solving Equations

The process of solving equations is much like uncovering clues in a detective story. Each transformation you apply brings you one step closer to revealing the identity of the unknown variable. The most fundamental principle in solving algebraic equations is the 'balance principle'—what you do to one side of the equation, you must do to the other to maintain the equality.

When you're looking to solve an equation like \(2x + 3 = 7\), the steps include isolating the variable on one side. You would subtract 3 from both sides, which gives you \(2x = 4\), and then divide both sides by 2, which reveals \(x = 2\). This simple act of maintaining the balance while simplifying the equation is the cornerstone of finding solutions in algebra. Throughout your journey with algebra, you'll encounter various types of equations, from linear to quadratic, each with its own set of strategies for finding solutions.

Algebraic Solutions

The destination of solving equations is finding algebraic solutions. It's the 'X marks the spot' on the treasure map of mathematics. A solution to an algebraic equation is any value(s) that, when substituted into the original equation, transforms it into a true statement.

In the context of equivalent equations, such as \(2x - 6 = 4\) and \(2x = 10\), the solution is a shared point, like a crossroad, where both equations meet. As seen in the example, both of these distinct-looking equations have the shared solution \(x = 5\).

• It becomes evident that while equations may look different, they can still share the same solution set; this is the beauty of algebraic solutions.

These solutions can be visualized on a number line or within a coordinate system, especially when dealing with equations involving two variables. Regardless of the methods used, arriving at the algebraic solution is a rewarding end that confirms your understanding and application of mathematical principles.