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Chapter 1: Problem 45

Solve each equation involving rational expressions. Identify each equation as an identity, an inconsistent equation, or a conditional equation. $$4+\frac{6}{y-3}=\frac{2 y}{y-3}$$

Short Answer

Expert verified
The equation is an inconsistent equation with no valid solution.

Step by step solution

01

Determine Common Denominator

Identify the common denominator for all rational expressions in the equation. The common denominator is \(y - 3\).
02

Multiply by Common Denominator

Multiply every term in the equation by the common denominator \(y - 3\) to eliminate the fractions. \[4(y-3) + 6 = 2y\]
03

Simplify the Equation

Distribute and simplify the terms: \[4y - 12 + 6 = 2y\]. This simplifies further to \[4y - 6 = 2y\]
04

Solve for y

Move all terms involving \( y \) to one side of the equation: \[4y - 2y = 6\]. This simplifies to \[2y = 6\]. Divide both sides by 2 to find \[ y = 3 \]
05

Check for Restrictions

Check if \( y = 3 \) creates any restrictions based on the original equation. Substituting \( y = 3 \) makes the denominator zero, which is undefined. Therefore, \( y = 3 \) is not a valid solution.
06

Identify the Type of Equation

Since there is no valid solution, the equation is inconsistent.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

common denominator
When dealing with rational expressions, it is crucial to work with a common denominator. A common denominator allows us to combine or compare fractions easily. In the given equation, we have two fractions: \(\frac{6}{y-3}\) and \(\frac{2 y}{y-3}\). The denominator for both fractions is the same, which is \(y-3\). This simplifies our task as we can multiply every term in the equation by this common denominator. By doing so, we eliminate the fractions and work with simpler algebraic expressions. Finding and using the correct common denominator is a critical first step.
simplify the equation
After determining the common denominator, the next step is to simplify the equation. By multiplying each term by the common denominator \(y-3\), we clear the fractions:
  • \[4(y-3) + 6 = 2y\]
.We then distribute and combine like terms to simplify the equation further:
  • \[4y - 12 + 6 = 2y\]
. This step-by-step simplification is essential, leading us to a more manageable equation: \[4y - 6 = 2y\]. Simplification helps us see the core algebraic structure of the expression.
inconsistent equation
An inconsistent equation is an equation that has no solution. In our exercise, after manipulating the equation, we ultimately find \( y = 3 \). However, substituting \( y = 3 \) back into the original equation reveals a problem. When \( y = 3 \), the denominator becomes zero, making the expression undefined. Because dividing by zero is impossible in mathematics, we conclude that there is no valid solution. Therefore, we identify this as an inconsistent equation. This means that no value of \( y \) will satisfy the original equation, and we officially declare it as having no solution.
solve for variable
Solving for a variable involves isolating the variable on one side of the equation. In our simplified equation, we have \[4y - 6 = 2y\]. To isolate \( y \), we move all terms involving \( y \) to one side:
  • \(4y - 2y = 6\), which simplifies to \(2y = 6\).
We then solve for \( y \) by dividing both sides by 2:
  • \(y = 3\).
However, remember to check if this solution creates any contradictions or undefined situations in the original equation.
undefined solution
An undefined solution occurs when solving the equation leads to a value that causes division by zero or other undefined mathematical operations. In this case, setting \( y = 3 \) leads to the term \( y-3 \) in the denominator, which becomes zero. Since dividing by zero is undefined, we must reject this solution. An undefined solution means that the proposed solution does not work within the context of the original equation, highlighting the importance of checking solutions in rational expressions carefully.

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