Problem 33
Question
Find and correct the error. $$ \begin{aligned} &\frac{x+2}{y+3}+\frac{y-4}{y+3}=\frac{(y+2)(y-4)}{(y+3)^{2}}\\\ &=\frac{y^{2}-2 y-8}{y^{2}+6 y+9} \end{aligned} $$
Step-by-Step Solution
Verified Answer
The correct equation after fixing the error should be \( \frac{x+2}{y+3}+\frac{y-4}{y+3} = \frac{(x+y-2)}{(y+3)} \), which further simplifies to \( \frac{x+y-2}{(y+3)} \).
1Step 1: Identify the Error
Initially, spot the mistake in the equation. Here, the error lies in the right-hand side of the very first equation where it's stated that \( \frac{x+2}{y+3}+\frac{y-4}{y+3}=\frac{(y+2)(y-4)}{(y+3)^{2}} \) . The numerator in the right-hand side does not match the sum of the numerators in the left-hand side.
2Step 2: Correct Factorization
Try to correctly factorize the expression. We get \( \frac{x+2}{y+3}+\frac{y-4}{y+3} = \frac{(x+y-2)}{(y+3)} \) , which simplifies to \( \frac{x+y-2}{y+3} \).
3Step 3: Simplify
On further simplifying this expression, we get \( \frac{x+y-2}{(y+3)} \) .
Key Concepts
Error Identification in Rational ExpressionsFactorization and Its ImportanceSimplification of Rational Expressions
Error Identification in Rational Expressions
Understanding how to identify errors in rational expressions is crucial for solving and simplifying these mathematical structures. Errors in expressions typically arise from miscalculation or misunderstanding of algebraic rules.
In many exercises, like the one we're focusing on, the error occurs in the addition or simplification of terms. For example, when given two fractions, adding them involves combining their numerators over a common denominator. If this rule isn't followed precisely, it results in errors.
In our case, the mistake happens when the numerators of the fractions \( \frac{x+2}{y+3} \) and \( \frac{y-4}{y+3} \) were incorrectly combined into a single product expression \( (y+2)(y-4) \). This misstep violates the basic rules for adding fractions because the numerators should be added directly rather than turning them into a product.
In many exercises, like the one we're focusing on, the error occurs in the addition or simplification of terms. For example, when given two fractions, adding them involves combining their numerators over a common denominator. If this rule isn't followed precisely, it results in errors.
In our case, the mistake happens when the numerators of the fractions \( \frac{x+2}{y+3} \) and \( \frac{y-4}{y+3} \) were incorrectly combined into a single product expression \( (y+2)(y-4) \). This misstep violates the basic rules for adding fractions because the numerators should be added directly rather than turning them into a product.
Factorization and Its Importance
Factorization is the process of breaking down expressions into products of simpler terms. It simplifies expressions and often reveals underlying patterns.
When dealing with fractions, especially in rational expressions, factorization helps identify common factors in either the numerators or denominators, making simplification possible. In our original problem, the expression on the left-hand side needed re-examination and correct factorization.
Instead of multiplying terms inside the numerator, correctly adding the numerators \( x+2 \) and \( y-4 \) yields \( x+y-2 \). The error was attempting to factorize the terms as if solving a multiplication problem, whereas in addition, such behavior leads to inaccurate results.
When dealing with fractions, especially in rational expressions, factorization helps identify common factors in either the numerators or denominators, making simplification possible. In our original problem, the expression on the left-hand side needed re-examination and correct factorization.
Instead of multiplying terms inside the numerator, correctly adding the numerators \( x+2 \) and \( y-4 \) yields \( x+y-2 \). The error was attempting to factorize the terms as if solving a multiplication problem, whereas in addition, such behavior leads to inaccurate results.
Simplification of Rational Expressions
Simplification is about reducing expressions to their simplest form, making them easier to handle and understand in further calculations.
Once errors are identified and proper terms are established, simplification follows naturally. For equations like \( \frac{x+2}{y+3} + \frac{y-4}{y+3} \), simplification involves combining the terms in the numerator to a single fraction \( \frac{x+y-2}{y+3} \).
It's important to ensure no common factors exist between the numerator \( x+y-2 \) and the denominator \( y+3 \), allowing for a true and fully simplified expression. Simplification not only clarifies the equation but also enhances computational efficiency.
Once errors are identified and proper terms are established, simplification follows naturally. For equations like \( \frac{x+2}{y+3} + \frac{y-4}{y+3} \), simplification involves combining the terms in the numerator to a single fraction \( \frac{x+y-2}{y+3} \).
It's important to ensure no common factors exist between the numerator \( x+y-2 \) and the denominator \( y+3 \), allowing for a true and fully simplified expression. Simplification not only clarifies the equation but also enhances computational efficiency.
Other exercises in this chapter
Problem 32
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View solution Problem 32
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In Exercises \(31-33,\) state whether the variables model direct variation, inverse variation, or neither. HOURS AND PAY RATE The number of hours \(h\) that you
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Factor first, then solve the equation. Check your solutions. \(\frac{2}{y-2}+\frac{1}{y+2}=\frac{4}{y^{2}-4}\)
View solution