- Identify the sum \(x+y = -9\) and product \(xy=20\) from the given equations.
- Form the quadratic equation \(t^2 + 9t + 20 = 0\) whose roots are \(x\) and \(y\).
- Factor the equation as \((t+4)(t+5) = 0\) to find the roots \(t = -4\) and \(t = -5\).
- Conclude that the other solution pair is \(x = -5, y = -4\).
- Final answer: \(x = \boxed{-5}, y = \boxed{-4}\)
Explanation
- Analyzing the System of Equations
We are given a system of two equations with two variables \(x\) and \(y\):
- \(xy = 20\)
- \(x + y + 9 = 0\)
From the second equation, we can write the sum of the variables as:
\(x + y = -9\)
We recognize that \(x\) and \(y\) are the roots of a quadratic equation of the form:
\(t^2 - (x+y)t + xy = 0\)
Substituting the sum \(x+y = -9\) and the product \(xy = 20\) into this form, we get:
\(t^2 - (-9)t + 20 = 0\)
\(t^2 + 9t + 20 = 0\)
- Solving for the Variables
To find the values of \(x\) and \(y\), we solve the quadratic equation \(t^2 + 9t + 20 = 0\). We can factor this expression by looking for two numbers that multiply to \(20\) and add up to \(9\). These numbers are \(4\) and \(5\).
Thus, the equation factors as:
\((t + 4)(t + 5) = 0\)
The roots of the equation are:
\(t_1 = -4\)
\(t_2 = -5\)
This means the solution pairs \((x, y)\) for the system are \((-4, -5)\) and \((-5, -4)\). Given that one solution is \(x = -4, y = -5\), the other solution set is \(x = -5, y = -4\).
Examples
In real-world applications, such as calculating the dimensions of a rectangular area with a known perimeter and area, we often encounter systems of equations like this. For instance, if you know the total length of fencing (perimeter) and the area of a garden, you can set up a system of equations to determine the length and width. Solving these systems is a fundamental skill for engineers and designers to ensure they optimize space effectively while adhering to physical constraints.