Problem 29 - College Algebra

Question

Solve each system by the method of your choice. $$\begin{aligned} &3 x^{2}+4 y^{2}=16\\\ &2 x^{2}-3 y^{2}=5 \end{aligned}$$

Step-by-Step Solution

Answer

The solutions to the system of equations are (2,1), (2, -1), (-2, 1) and (-2, -1)

Step 1: Rearrange the first equation

With the first equation \(3x^2+4y^2=16\), we can rearrange it to express it in terms of \(x^2\), \( x^2= \frac{16-4y^2}{3}\)

Step 2: Apply substitution

Replace \(x^2\) in the second equation \(2x^2-3y^2=5\) with \( \frac{16-4y^2}{3}\) from step 1. This gives us \(2(\frac{16-4y^2}{3}) - 3y^2=5\). This is a linear equation in the variable \(y\).

Step 3: Solve for y

Solving the equation from Step 2, we find \( y=\sqrt{1}\) or \( y=-\sqrt{1}\) which simplifies to \( y=1\) or \( y=-1\).

Step 4: Substitute y-values in equation

Substitute these values of \( y \) in equation 1 obtained in step 1 to get the corresponding \( x \) values, i.e., \(x^2 = \frac{16-4(1^2)}{3}\) or \(x^2 = \frac{16-4((-1)^2)}{3}\). This simplifies to \( x=\sqrt{4}\) or \(x= -\sqrt{4}\) for \(y = 1\) and \( x=\sqrt{4}\) or \( x= -\sqrt{4}\) for \(y = -1\). This finally gives us solutions \( x=2, y=1\) or \( x=2, y=-1\) or \( x=-2, y=1\) or \( x=-2, y=-1\).

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation where the highest power of the variable is squared (raised to the power of 2). In general form, a quadratic equation can be written as \(ax^2 + bx + c = 0\), where \(a, b, \) and \(c\) are constants and \(a eq 0\). In the context of our system of equations, the quadratic terms involve the variables \(x^2\) and \(y^2\) as seen in both equations provided:

• \(3x^2 + 4y^2 = 16\)
• \(2x^2 - 3y^2 = 5\)

Using quadratic equations, we explore how the unknowns \(x\) and \(y\) interact when squared and combined with constants. Quadratic equations are part of system solving when more than one unknown is involved, leading us to methods like substitution and simultaneous resolution to find values that satisfy both equations. Understanding the basic form, coefficients, and constants within these equations is crucial for further steps in solving them. Applying techniques to manipulate one equation often makes solving its system more approachable.

Substitution Method

The substitution method is a strategy used to solve systems of equations where you solve one equation for one variable and then substitute that expression into the other equation. This allows you to eliminate one of the variables and focus on solving for the remaining one. Here's how it works in our given problem:1. **Rearrange Equations**: We started with the equation \(3x^2 + 4y^2 = 16\) and rearranged it to solve for \(x^2\). This gave us \(x^2 = \frac{16 - 4y^2}{3}\).2. **Substitute**: Next, this expression for \(x^2\) was substituted into the second equation \(2x^2 - 3y^2 = 5\). This step changes the system to focus solely on the variable \(y\), forming the equation \(2(\frac{16 - 4y^2}{3}) - 3y^2 = 5\).Substituting one equation into another transforms the system into a readily solvable form, ideally a single-variable equation. It simplifies the problem and reduces it to basic algebra, making it a highly effective method when dealing with two equations involving more than one variable. This method is particularly useful in our exercise to streamline solving complex systems like quadratic equations.

Solving Equations

Solving equations, particularly in a system, means finding the values of variables that satisfy all given constraints. Once equations are manipulated (as seen with substitution), they are often solved step-by-step:- **Solve for One Variable**: In our exercise, after substitution, the main focus was on \(y\) resulting in \(y = 1\) or \(y = -1\). Solving for this variable simplified the complex equation.- **Back-Substitute**: After finding \(y\), we back-substitute these values into one of our rearranged equations to solve for \(x\). This routine step ensures all solutions are verified against both initial equations.The key to solving equations is a methodical approach, breaking down complex relationships into simpler tasks. Each step reduces the complexity until you isolate the variables, leading to solutions like \(x = 2, y = 1\) or \(x = -2, y = -1\), which resolve the original system. Mastery of solving systems requires practice in identifying the most effective sequence to isolate and solve variables within given constraints.