Problem 42 - Intermediate Algebra

Question

Solve each system of equations for real values of \(x\) and \(y.\) $$ \left\\{\begin{array}{l} 2 x^{2}-y^{2}+2=0 \\ 3 x^{2}-2 y^{2}+5=0 \end{array}\right. $$

Step-by-Step Solution

Answer

The solutions are \((1, 2)\), \((1, -2)\), \((-1, 2)\), and \((-1, -2)\).

Step 1: Express Both Equations

Start by writing the system of equations individually: \( 2x^2 - y^2 + 2 = 0 \) and \( 3x^2 - 2y^2 + 5 = 0 \).

Step 2: Rearrange Equations for Simplicity

Rearrange each equation to solve for \( y^2 \). From the first equation, we have \( y^2 = 2x^2 + 2 \). From the second equation, \( y^2 = \frac{3}{2}x^2 + \frac{5}{2} \).

Step 3: Set Equations Equal

Since both are equal to \( y^2 \), set them equal: \( 2x^2 + 2 = \frac{3}{2}x^2 + \frac{5}{2} \).

Step 4: Solve for \( x^2 \)

Rearrange to solve for \( x^2 \):\[ 2x^2 + 2 = \frac{3}{2}x^2 + \frac{5}{2} \]Multiply every term by 2 to eliminate the fraction:\[ 4x^2 + 4 = 3x^2 + 5 \]Subtract \( 3x^2 \) and 4 from both sides:\[ x^2 = 1 \].

Step 5: Find Values of \( x \)

Solve for \( x \) by taking the square root: \( x = \pm 1 \).

Step 6: Find Corresponding \( y \) Values

Substitute \( x = 1 \) and \( x = -1 \) in \( y^2 = 2x^2 + 2 \):- For \( x = 1 \), \( y^2 = 2(1)^2 + 2 = 4 \), so \( y = \pm 2 \).- For \( x = -1 \), \( y^2 = 2(-1)^2 + 2 = 4 \), so \( y = \pm 2 \).

Step 7: List All Solutions

The solutions to the system are \((1, 2)\), \((1, -2)\), \((-1, 2)\), and \((-1, -2)\).

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra and mathematics overall. They take the shape of a polynomial equation in which the highest degree is two. The general form of a quadratic equation is given by:

• \( ax^2 + bx + c = 0 \)

Here, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). Quadratic equations can have two, one, or zero real solutions. Everything hinges on the discriminant \( b^2 - 4ac \):

• If the discriminant is positive, there are two distinct real solutions.
• If it is zero, there's exactly one (repeated) real solution.
• If negative, the solutions are complex and not real.

Quadratics can be used in a variety of problems, like calculating distances, velocities, or growth rates, just to name a few. Solving them often involves finding square roots, which requires careful mathematical manipulation to ensure no steps are missed.

Algebraic Manipulation

Algebraic manipulation is key in simplifying and solving equations, especially systems of equations. It involves using various algebraic rules to transform expressions into a more manageable form. For example, transforming an equation to isolate a variable can involve:

• Adding or subtracting terms from both sides of an equation.
• Multiplying or dividing both sides by a constant to simplify fractions.
• Factoring expressions to reveal hidden solutions.

In our problem, algebraic manipulation was used to simplify and compare \( y^2 \) values derived from two original equations. By expressing both equations in terms of \( y^2 \), setting them equal, and eliminating fractions, we traveled through a precise journey to isolate \( x^2 \). This ordered and logical manipulation ensures that equations remain balanced and solutions remain accurate.

Simultaneous Equations

Simultaneous equations are sets of equations that are solved together, interacting to find common variable values. They can be linear or include nonlinear terms, as in our example. When solving simultaneous equations, one method is substitution, where:

• One equation is manipulated to express a variable in terms of the other.
• This expression is substituted into the second equation, reducing the system to a single equation with one variable.
• After solving, substitute back to find the remaining variables.

Another powerful method is elimination, which our exercise leaned on:

• Manipulating or combining equations to eliminate variables step-by-step.
• This method often requires tracking multiple terms that disappear through addition or subtraction.

Understanding simultaneous equations involves linking various algebraic and numeric concepts into a coherent strategy for finding solutions that satisfy all given conditions. They showcase the beauty of algebra in resolving complex problems with logical precision.