Problem 28 - Precalculus: Mathematical for Calculus

Question

Find all solutions of the system of equations. $$\left\\{\begin{aligned}x y &=24 \\\2 x^{2}-y^{2}+4 &=0\end{aligned}\right.$$

Step-by-Step Solution

Answer

Solutions are (4, 6) and (-4, -6).

Step 1: Solve First Equation for y

The first equation given is \( xy = 24 \). To solve for \( y \), divide both sides by \( x \): \( y = \frac{24}{x} \). This equation expresses \( y \) in terms of \( x \).

Step 2: Substitute y in Second Equation

The second equation is \( 2x^2 - y^2 + 4 = 0 \). Substitute \( y = \frac{24}{x} \) into this equation: \[ 2x^2 - \left(\frac{24}{x}\right)^2 + 4 = 0. \]

Step 3: Simplify and Create Quadratic Equation

Simplify the substituted equation: \[ 2x^2 - \frac{576}{x^2} + 4 = 0. \] Multiply through by \( x^2 \) to eliminate the fraction: \[ 2x^4 + 4x^2 - 576 = 0. \] Define a substitution \( z = x^2 \), converting the equation to \( 2z^2 + 4z - 576 = 0 \).

Step 4: Solve the Quadratic Equation

Now solve the quadratic equation \( 2z^2 + 4z - 576 = 0 \) using the quadratic formula, \( z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = 4 \), \( c = -576 \). First, compute the discriminant: \( b^2 - 4ac = 4^2 - 4 \times 2 \times (-576) = 4624 \). Calculate \( z \): \[ z = \frac{-4 \pm \sqrt{4624}}{4}. \] Since \( \sqrt{4624} = 68 \), \[ z = \frac{-4 \pm 68}{4}. \] This gives solutions \( z = 16 \) and \( z = -18 \). Since \( z = x^2 \) and cannot be negative, \( x^2 = 16 \).

Step 5: Find Possible x Values

From \( x^2 = 16 \), the possible values for \( x \) are \( x = 4 \) and \( x = -4 \).

Step 6: Calculate Corresponding y Values

Using the values for \( x \), calculate \( y \) from \( y = \frac{24}{x} \): - If \( x = 4 \), then \( y = \frac{24}{4} = 6 \).- If \( x = -4 \), then \( y = \frac{24}{-4} = -6 \).

Step 7: Enumerate Solution Pairs

The pairs \( (x, y) \) that satisfy both equations are \( (4, 6) \) and \( (-4, -6) \).

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental aspect of algebra and can be recognized by their highest degree of a variable being squared. The general form of a quadratic equation is expressed as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero.

These equations often arise when dealing with systems of equations, as seen in our exercise. When manipulating the equations or substituting values, transforming them into a quadratic form can be key to solving them.

In the given exercise, during Step 3, we derive a quadratic equation \( 2z^2 + 4z - 576 = 0 \) by substituting \( z = x^2 \). This transformation simplifies the solving process and showcases how quadratic equations can be derived and solved to find solutions to more complex systems.

Substitution Method

The substitution method is a powerful technique used to solve systems of equations, particularly useful when one of the equations can be easily solved for one of the variables. It involves two main steps:

• First, solve one of the equations for one variable in terms of the other.
• Next, substitute this expression into the second equation, effectively reducing the system to a single equation in one variable.

In our example, we start with the equation \( xy = 24 \) to solve for \( y \), giving us \( y = \frac{24}{x} \).

Then, we substitute this expression for \( y \) into the second equation \( 2x^2 - y^2 + 4 = 0 \). This substitution transforms the equation into a form that we can further manipulate and solve. The substitution step is crucial for lessening the complexity of solving systems of equations, enabling focus on one variable at a time.

This method exemplifies the strength of substitution in simplifying multi-variable problems.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations to solve for unknowns. It includes expanding, factoring, and combining like terms. Mastery of these skills is essential when working with systems of equations.

In our exercise, after substituting \( y = \frac{24}{x} \) into the second equation, algebraic manipulation helps us convert \( 2x^2 - \frac{576}{x^2} + 4 = 0 \) into a quadratic form by eliminating the fraction. We multiply through by \( x^2 \):

• Expanding fractions: Manage terms to avoid dealing with fractions, which can complicate the solving process.
• Factoring or completing the square: These techniques can then be used to solve the resulting quadratic equation efficiently.

This refinement enables us to work with simpler expressions, leading to the solution. Effective algebraic manipulation brings clarity and simplicity to complex systems, underscoring its importance in solving equations.