Problem 86 - Introductory Algebra for College Students

Question

Will help you prepare for the material covered in the next section. In each exercise, solve the given equation. $$(5 x-1)+1=5 x+5$$

Step-by-Step Solution

Answer

The equation does not have any solutions.

Step 1: Simplify the Equation

We start by simplifying the equation, \(5x -1 + 1 = 5x +5\), which simplifies to \(5x = 5x + 5\).

Step 2: Isolate the Variable on One Side of the Equation

Subtract \(5x\) from both sides of the equation to get the value of \(x\). This results in the equation 0 = 5.

Step 3: Solving for 'x'

We can see that this leads to a statement that is not true (0 = 5). Hence, there are no solutions for this equation.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that involve expressions with variables, constants, and sometimes operators. These equations balance two expressions, which often include numbers and variables.

• Variables: Symbols representing unknown values, usually denoted by letters like \(x, y, z\).
• Constants: Fixed values, such as numbers.
• The goal is to find the values of variables that make the equation true.

To solve algebraic equations, one typically performs operations to isolate the variable on one side. It usually involves steps like addition, subtraction, multiplication, or division. In the exercise, the equation given is \((5x - 1) + 1 = 5x + 5\). This equation is balanced, meaning the expressions on both sides are equal when a specific value is substituted for the variable, \(x\). Learning to solve these equations is foundational, as it builds skills necessary for more complex algebraic manipulations.

No Solution Equations

Certain equations lead to statements that are impossible or false, indicating that no value will satisfy the equation. These are known as no solution equations.In the given exercise, following simplification, we reach the equation \(0 = 5\), which is evidently a false statement. False statements result in equations having no possible solutions because no value for the variable will make both sides equal.Steps to identify no solution equations:

• Simplify the equation as much as possible, reducing any like terms.
• Attempt standard operations to isolate variables.
• Evaluate the result of these operations; if they lead to a contradiction, such as \(0 = 5\), then the equation has no solution.

Understanding no solution scenarios is critical for recognizing the limitations of mathematical models and ensuring the choice of appropriate methods in problem-solving.

Equation Simplification

Simplification is an important step in solving algebraic equations. It involves rewriting the equation in its simplest form to make it easier to solve. Equation simplification often includes combining like terms, reducing fractions, or eliminating unnecessary elements. In our exercise, the initial step shown is to simplify \((5x - 1) + 1 = 5x + 5\) to \(5x = 5x + 5\).Simplifying helps in identifying how terms relate to each other and finding a path to solve them.Here are key strategies for simplifying equations:

• Combine like terms, such as numbers or similar variable terms, to reduce complexity.
• Apply the distributive property when necessary to handle expressions involving sums and products.
• Remain vigilant for opportunities to cancel terms or simplify fractions.

By simplifying the initial equation, it becomes apparent that further basic operations (like subtracting \(5x\)) results in \(0 = 5\), signaling a contradiction and highlighting that the equation has no solution. This step is crucial, as simplifying too early or incorrectly might conceal the nature of the solution.