- The system of equations \(xy=12\) and \(x+y=7\) corresponds to the quadratic equation \(t^2 - 7t + 12 = 0\).
- Factoring the quadratic equation gives \((t-4)(t-3) = 0\), with roots \(t=4\) and \(t=3\).
- Since one solution set is \(x=4, y=3\), the other solution set is obtained by swapping the variables.
- The other solution set is \(x=3, y=4\).
- Final Answer: \(\boxed{x=3, y=4}\)
Explanation
- Problem Analysis and Theoretical Background
We are given a system of two equations with two variables, \(x\) and \(y\):
\[xy = 12\]
\[x + y = 7\]
We are told that one solution set is \(x = 4, y = 3\). We need to find the other possible solution set for this system. A well-known property of quadratic equations is that if \(x\) and \(y\) are roots of a quadratic equation in \(t\), then that equation can be written as \(t^2 - (x+y)t + xy = 0\). This is derived from Vieta's formulas, which relate the sum and product of the roots to the coefficients of the polynomial.
- Solving the Quadratic Equation
Using the given values for the sum and product of the variables, we can construct the quadratic equation:
\[t^2 - (x+y)t + xy = 0\]
Substituting the known values \(x+y = 7\) and \(xy = 12\), we get:
\[t^2 - 7t + 12 = 0\]
To find the possible values for \(x\) and \(y\), we solve this quadratic equation for \(t\). We can factor the quadratic expression by looking for two numbers that multiply to \(12\) and add up to \(-7\). These numbers are \(-4\) and \(-3\).
Thus, the equation can be factored as:
\[(t - 4)(t - 3) = 0\]
- Finding the Other Solution Set
The roots of the equation \((t - 4)(t - 3) = 0\) are \(t = 4\) and \(t = 3\). This tells us that any solution pair \((x, y)\) for the system must consist of the values \(4\) and \(3\).
We were given the solution set \(x = 4, y = 3\). By swapping these values, we obtain the other solution set:
\[x = 3, y = 4\]
We can easily verify this:
\(x \times y = 3 \times 4 = 12\)
\(x + y = 3 + 4 = 7\)
Both equations are satisfied. Therefore, the other solution set is \(x = 3, y = 4\).
Examples
Systems of equations like these are fundamental in various fields, such as physics and economics, where you often need to determine two unknown quantities given their sum and product. For instance, if you are designing a rectangular garden with a fixed area of 12 square meters and a perimeter that requires a sum of sides (length + width) of 7 meters, you can solve the system to find the dimensions. Understanding how to use Vieta's formulas makes solving these problems efficient and systematic, ensuring you find all possible dimensions for your design.