Problem 59 - College Algebra and Calculus: An Applied Approach

Question

What is meant by "equivalent equations"? Give an example of two equivalent equations.

Step-by-Step Solution

Answer

Equivalent equations are equations which have the same solutions. The equations \(2x + 3 = 7\) and \(2x = 7 - 3\) are equivalent as both solutions yield \(x = 2\).

Step 1: Definition

Equivalent Equations are equations that have the same solutions. This essentially means if you solve both equations, you will get the same answers.

Step 2: Providing Example

Let's consider the equations \(2x + 3 = 7\) and \(2x = 7 - 3\). Here, both equations are equivalent because they have the same solution, which is \( x = 2 \).

Key Concepts

Algebraic Equations

In algebra, an equation is like a balance scale. Just as adding or removing the same weight from both sides of a scale keeps it balanced, you can perform the same operations on both sides of an equation without changing its solution set. Algebraic equations come in various forms, from simple linear equations like (2x + 3 = 7), to more complex polynomial, exponential, and rational equations.
Algebraic equations are fundamental tools used to describe relationships between quantities and find unknown values. Understanding how to manipulate these equations is crucial in solving mathematical and real-world problems. For example, if you have the amount you spent on lunch (\(7) and you know the cost per item (\)2 plus a fixed tax of $3), you could use an algebraic equation to find out the number of items you purchased.

Solving Equations

Solving equations is a central task in algebra. To find the solutions of an equation, we apply various mathematical operations to isolate the variable. For instance, with the equation (2x + 3 = 7), we aim to figure out the value of (x) that makes the equation true. We do this by reversing the operations that have been applied to (x). This means we subtract (3) from both sides to get rid of the constant term, giving us a simpler equation (2x = 4). Dividing both sides by (2), we find that (x = 2), which is the solution.

• Identify the operations performed on the variable
• Reverse these operations in the reverse order they were applied
• Check the solution by substituting it back into the original equation

By mastering these steps, students can solve increasingly complex equations, building up from linear equations to those involving multiple variables or higher powers.

Algebra Fundamentals

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. The fundamental principles of algebra involve operations such as addition, subtraction, multiplication, and division, as well as the use of variables such as (x) and (y) to represent unknown values. An understanding of these principles is essential to progress in algebra.
In the example provided, the operation involving (2x + 3 = 7), guides us to find the value of (x). By following algebraic principles, we simplify and rearrange the equation to isolate the variable and solve it. A deep comprehension of these basics enables students to manipulate and solve not just 'equivalent equations' but a vast array of algebraic expressions and models, paving the way for more advanced mathematical exploration.