Problem 45 - Elementary Algebra

Question

For the following problems, solve each conditional equation. If the equation is not conditional, identify it as an identity or a contradiction. $$ -4(5 y+3)+5(1+4 y)=0 $$

Step-by-Step Solution

Answer

Question: Determine whether the linear equation -4(5y + 3) + 5(1 + 4y) = 0 is an identity, a contradiction, or a conditional equation. Answer: The given linear equation is a contradiction.

Step 1: Simplify the equation

First, let's distribute the terms within each parentheses and then simplify the equation: $$ -4(5y + 3) + 5(1 + 4y) = 0 $$ $$ -20y - 12 + 5 + 20y = 0 $$ Now, combine like terms: $$ -20y + 20y - 12 + 5 = 0 $$

Step 2: Determine if it's an identity, contradiction, or conditional equation

Now, we can see that -20y and +20y cancel each other out, leaving the equation: $$ -7 = 0 $$ This equation is a contradiction since -7 is not equal to 0. There are no solutions for y, and the given equation is not conditional.

Key Concepts

Identity Equations

An identity equation is one where both sides are equal for all values of the variable(s) involved. This means the equation holds true regardless of the number you plug in for the variable. For instance, consider the equation \( x - 2 = x - 2 \).

• This equation will always be true, as both sides are identical once simplified.
• Every number you substitute for \( x \) will satisfy the equation.
• If simplifying an equation results in a tautology like \( 0 = 0 \), it is confirmed as an identity.

Identity equations provide a broad perspective on equivalence and can handle infinite sets of solutions.

Contradiction Equations

Contradiction equations are aptly named because they lead to statements that are never true. Such equations hold no possible solutions because the statements they create are logically inconsistent. For example, if you simplify an equation and end with something like \( -7 = 0 \), like in our exercise, it reveals a contradiction.

• No value of the variable will ever satisfy the condition.
• These equations often indicate an error if a solution was expected.
• They are also useful as they show limitations in certain assumptions or calculations.

Understanding contradictions can help students recognize unnecessary errors or infeasible solutions.

Algebraic Simplification

Algebraic simplification is a critical skill when working with equations, involving the reduction of equations to a simpler form for easier analysis. This usually involves a few key steps:

• Expanding and distributing terms: Such as turning \(-4(5y + 3)\) into \(-20y - 12\).
• Combining like terms: Merging terms that have the same variables and exponents, like \(-20y + 20y\).
• Reducing constants and coefficients to make further steps straightforward.

These steps not only clarify complex expressions but also reveal key characteristics of the equation, like whether it's a contradiction or an identity.

Equation Solving Methods

Equation solving involves finding variable values that satisfy a given equation. Though the specific steps may vary based on the equation type, general methods can be applied to streamline the process:

• Identify the equation type: Is it conditional, a contradiction, or an identity?
• Re-structure the equation to isolate the variable. This may involve moving terms to different sides.
• Use algebraic operations, such as addition, subtraction, multiplication, or division, to simplify the equation further.
• Verify the solution, ensuring it satisfies the original equation if possible.

These methods are foundational, providing a framework for solving equations effectively across various algebra levels.