Verified Answer :
- Identify the values of the variables: Given \(x + y = 10\) and \(x = 6\), calculate \(y = 4\).
- Evaluate the numerator: Calculate \(|x - 3y| = |6 - 3(4)| = 6\).
- Evaluate the denominator: Calculate \(|xy| = |6 \times 4| = 24\).
- Calculate the final expression: Divide the numerator by the denominator to get \(\frac{6}{24} = \frac{1}{4}\).
- The final answer is \(\boxed{\frac{1}{4}}\).
Explanation
- Problem Analysis and Variables
We are given two equations involving two variables \(x\) and \(y\):
- \(x + y = 10\)
- \(x = 6\)
We want to evaluate the expression \(\frac{|x - 3y|}{|xy|}\). Our strategy is to first solve for the variable \(y\) using the given equations, then substitute both \(x\) and \(y\) into the expression to find its numerical value.
- Solving for the Variables
Using the equation \(x + y = 10\) and substituting the known value \(x = 6\), we have:
\[6 + y = 10\]
Subtracting \(6\) from both sides gives:
\[y = 10 - 6 = 4\]
Thus, the values for our variables are \(x = 6\) and \(y = 4\).
- Evaluating the Expression
Now we evaluate the numerator \(|x - 3y|\) by substituting \(x = 6\) and \(y = 4\):
\[|x - 3y| = |6 - 3(4)| = |6 - 12| = |-6|\]
Since the absolute value of \(-6\) is \(6\), the numerator is \(6\).
Next, we evaluate the denominator \(|xy|\):
\[|xy| = |6 \times 4| = |24| = 24\]
The denominator is \(24\).
- Final Calculation
Finally, we calculate the quotient of the numerator and the denominator:
\[\frac{|x - 3y|}{|xy|} = \frac{6}{24}\]
By simplifying the fraction \(\frac{6}{24}\), we divide both the numerator and the denominator by their greatest common divisor, which is \(6\):
\[\frac{6 \div 6}{24 \div 6} = \frac{1}{4}\]
The value of the expression is \(\frac{1}{4}\), which is equal to \(0.25\). Thus, the final result is \(\boxed{\frac{1}{4}}\).
Examples
Suppose you are managing a simple budget where the total amount spent on two categories, A and B, must be 10 units. If you know that the amount spent on category A is exactly 6 units, you can easily determine the amount spent on category B as 4 units. If you need to evaluate a specific relationship between these costs, such as the difference between category A and three times category B relative to their product, you can use substitution to calculate that specific ratio, helping you analyze the proportionality of your expenses.