Problem 32 - Algebra and Trigonometry

Question

Solve each system by the method of your choice. $$ \left\\{\begin{array}{l} {x^{2}+4 y^{2}=20} \\ {x y=4} \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x=2, y=2\) and \(x=-2, y=-2\).

Step 1: Rearrange the linear equation

Rearrange the equation \(xy = 4\) to express one variable in terms of the other variable. We get \(x = \frac{4}{y}\) or \(y = \frac{4}{x}\).

Step 2: Substitute in the quadratic equation.

Substitute \(x = \frac{4}{y}\) into the quadratic equation \(x^{2}+4y^{2}=20\). This gives \((\frac{4}{y})^2+4y^{2}=20\) which simplifies to \(16+y^{2}=5y^{2}\). Solving for \(y\) gives \(y^{2}=4\), so \(y=2\) or \(y=-2\).

Step 3: Substitute \(y\) back into the linear equation

Substitute \(y=2\) into the linear equation \(x=\frac{4}{y}\) to obtain \(x=2\). Similarly, substitute \(y=-2\) into the linear equation to find \(x=-2\).

Key Concepts

Substitution Method

Understanding the substitution method is crucial for solving systems of nonlinear equations. It allows us to reduce a system of equations to a single variable, making it easier to solve. Here's how it works:

• Select one equation and solve for one variable in terms of the others.
• Substitute this expression into the other equations.
• Solve the resulting equation, which now has only one variable.
• Back-substitute the found value(s) into the original expression to find the other variables.

In our exercise, the linear equation is manipulated to express one variable in terms of another. When we substitute this expression into the quadratic equation, we reduce the system to a single variable, making it possible to find the solution(s) for that variable. It's a methodical process that requires careful algebraic manipulation, but once mastered, it becomes a powerful tool in the mathematician's toolbox.

Quadratic Equations

Quadratic equations are fundamental in algebra and often appear in various fields of science and engineering. A quadratic equation is typically written in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a \eq 0\). Solving a quadratic equation involves finding the values of \(x\) that satisfy the equation.

• One can solve quadratic equations by factoring, completing the square, using the quadratic formula, or graphing.
• In our context, after the substitution, the quadratic equation simplifies to a form that can be solved by isolating the variable.
• Recognizing that \(y^{2} = 4\) is a quadratic equation, we solve it by taking square roots to find that \(y = 2\) or \(y = -2\).

Students must get comfortable with recognizing and solving quadratic equations as they are a cornerstone in understanding more complex algebraic concepts.

Algebraic Manipulation

Algebraic manipulation encompasses a variety of techniques used to solve equations and simplify expressions. These skills are vital for solving systems of equations, especially nonlinear systems.

• Key techniques include expanding brackets, isolating variables, factorizing expressions, and working with exponents.
• In the provided solution, we manipulate the linear equation to isolate \(x\), which simplifies the substitution.
• We then apply algebraic manipulation to the quadratic equation to collect like terms and isolate \(y^{2}\), allowing us to solve for \(y\).
• Finally, we use back-substitution to solve for \(x\) using the values obtained for \(y\).

These manipulation techniques are not just about mechanical application but also about strategic thinking: choosing which variable to isolate, which equation to substitute into, and how to simplify to reach a solution efficiently.