- Substitute the coordinates of each point \((x, y)\) into the expression \(3x - 4y\).
- Compare the result of the substitution with the constant \(21\).
- Identify that \((-1, -6)\), \((7, 0)\), and \((11, 3)\) satisfy the equality \(3x - 4y = 21\).
- Conclude that \((-3, 3)\) is not a solution because it evaluates to \(-21\).
- The correct pairs are: \(\boxed{(-1, -6), (7, 0), (11, 3)}\).
Explanation
- Problem Analysis and Methodology
To determine which of the given ordered pairs \((x, y)\) are solutions to the linear equation \(3x - 4y = 21\), we substitute the \(x\) and \(y\) values of each coordinate pair into the left-hand side of the equation. An ordered pair is a solution if, after substitution, the resulting value equals the right-hand side, which is \(21\). The candidate pairs to evaluate are \((-1, -6)\), \((7, 0)\), \((11, 3)\), and \((-3, 3)\).
- Evaluation of Candidate Pairs
We test each candidate pair:
- For \((-1, -6)\):
\(3(-1) - 4(-6) = -3 + 24 = 21\).
Since this equals 21, the pair \((-1, -6)\) is a solution.
- For \((7, 0)\):
\(3(7) - 4(0) = 21 - 0 = 21\).
Since this equals 21, the pair \((7, 0)\) is a solution.
- For \((11, 3)\):
\(3(11) - 4(3) = 33 - 12 = 21\).
Since this equals 21, the pair \((11, 3)\) is a solution.
- For \((-3, 3)\):
\(3(-3) - 4(3) = -9 - 12 = -21\).
Since this does not equal 21, the pair \((-3, 3)\) is not a solution.
- Final Conclusion
Based on our systematic evaluation, the ordered pairs that satisfy the equation \(3x - 4y = 21\) are \((-1, -6)\), \((7, 0)\), and \((11, 3)\). The pair \((-3, 3)\) results in \(-21\), which does not satisfy the given condition. Thus, the solutions are \((-1, -6)\), \((7, 0)\), and \((11, 3)\).
Examples
Linear equations are foundational in fields like economics to represent budget constraints. For instance, if you have a budget to spend on two types of goods, \(x\) and \(y\), their prices per unit can be represented as coefficients in an equation like \(3x + 4y = 21\), where \(21\) is the total budget. Checking coordinate pairs helps identify all possible combinations of goods that use your exact budget, ensuring you don't overspend or waste resources. Similarly, this logic applies to any situation involving fixed constraints where two variables must balance to reach a target value.