- Test the first pair \((3, -8)\) by substituting \(x=3\) and \(y=-8\) into the equation \(3x - \frac{y}{4} = 11\), which results in \(11 = 11\).
- Test the second pair \((4, 4)\) by substituting \(x=4\) and \(y=4\) into the equation, which also results in \(11 = 11\).
- Conclude that since both pairs satisfy the equation, both are solutions.
- Final Answer: \(\boxed{\text{Both ordered pairs are solutions.}}\)
Explanation
- Problem Analysis
To determine which of the given ordered pairs, \((3, -8)\) and \((4, 4)\), satisfy the equation \(3x - \frac{y}{4} = 11\), we must substitute the \(x\) and \(y\) coordinates of each pair into the equation and verify if the left-hand side evaluates to 11. An ordered pair \((x, y)\) is a solution if the substitution results in a true statement.
- Testing the First Pair \((3, -8)\)
First, we test the ordered pair \((3, -8)\). Here, \(x = 3\) and \(y = -8\).
Substituting these into the equation:
\[3(3) - \frac{-8}{4} = 9 - (-2) = 9 + 2 = 11.\]
Since the evaluation equals 11, the ordered pair \((3, -8)\) is a solution to the equation.
- Testing the Second Pair \((4, 4)\)
Next, we test the ordered pair \((4, 4)\). Here, \(x = 4\) and \(y = 4\).
Substituting these into the equation:
\[3(4) - \frac{4}{4} = 12 - 1 = 11.\]
Since the evaluation equals 11, the ordered pair \((4, 4)\) is also a solution to the equation.
- Conclusion
Both tests resulted in 11, which matches the right-hand side of the given equation. Therefore, both ordered pairs \((3, -8)\) and \((4, 4)\) are solutions. Thus, the true statement is that both ordered pairs are solutions to the equation. The final answer is \(\boxed{\text{Both ordered pairs are solutions.}}\)
Examples
This concept is used in many real-world applications, such as quality control in manufacturing. If a company defines a specific production standard by an equation (like a tolerance range for a part's dimensions), they can check if different samples (represented as pairs of measurements) meet that standard. By substituting the dimensions of a sample into the equation, they can determine if the part is acceptable (a solution) or needs to be discarded.