Problem 45 - College Algebra Essentials

Question

let \(x\) represent one number and let y represent the other number. Use the given conditions to write a system of nonlinear equations. Solve the system and find the numbers. The difference between the squares of two numbers is \(3 .\) Twice the square of the first number increased by the square of the second number is 9. Find the numbers.

Step-by-Step Solution

Answer

The solution set to the system of equations is { (2,1), (2,-1), (-2,1), (-2,-1) }.

Step 1: Formulate the equations

Begin by translating the problem into two equations: 1) \(x^2 - y^2 = 3\) and 2) \(2x^2 + y^2 = 9\). These two equations are created based on the given conditions.

Step 2: Rearrange the equations

Rearrange each equation to isolate \(y^2\). For equation 1, \(y^2 = x^2 - 3\). For equation 2, \(y^2 = 9 - 2x^2\).

Step 3: Solving the system

As both equations equal to \(y^2\), set them equal to each other: \(x^2 - 3 = 9 - 2x^2\). Solving for x, we find \(x = 2\) or \(x = -2\).

Step 4: Solve for y

Substitute these values into one of the original equations to find the corresponding y value: if \(x = 2\) then \(y = 1\) or \(y = -1\) and if \(x = -2\) then \(y = 1\) or \(y = -1\).

Step 5: Check the solutions

Substitute \(x = 2, y = 1\) and \(x = 2, y = -1\) as well as \(x = -2, y = 1\) and \(x = -2, y = -1\) into the original equations to verify that all four pairs of solutions satisfy both equations.

Key Concepts

Square of a Number

When we talk about the square of a number, we're referring to multiplying that number by itself. For example, the square of 3 is calculated as \(3 \times 3\), resulting in 9. This is expressed in mathematical terms as \(3^2 = 9\). Squares have special properties such as:

• They are always non-negative, meaning the square of any number is either positive or zero.
• The square of a negative number is positive; for instance, \((-3)^2 = 9\).

In our exercise, we deal with two numbers, \(x\) and \(y\), and their squares to form different equations. These equations help us to find solutions by setting conditions based on relationships between their squares. Understanding squares is crucial here as it helps us simplify and solve equations.

Solving Equations

Solving equations involves finding the values for the variables that make the equation true. In the context of a system of nonlinear equations, such as:

• \(x^2 - y^2 = 3\)
• \(2x^2 + y^2 = 9\)

We aim to find values of \(x\) and \(y\) that work for both equations simultaneously. Here’s how you can approach it:

• Rearrange the equations to isolate a variable. In our case, we isolate \(y^2\) in both equations.
• Equate the isolated expressions to simplify the equations into one variable.
• Find the variable’s possible values and then substitute back to find the other variable’s values.

It's essential to check the solutions by placing them back into the original equations to verify that they satisfy both. That way, you ensure the complete accuracy of the solutions derived through calculations.

Substitution Method

The substitution method is a popular technique for solving systems of equations, especially when dealing with non-linear systems. The method involves solving one equation for one variable and then substituting that expression into the other equation(s). Here's how we applied it in our exercise:- We initially expressed both equations in terms of \(y^2\) (i.e., \(y^2 = x^2 - 3\) and \(y^2 = 9 - 2x^2\)).- By equating the two expressions for \(y^2\), we eliminated \(y\) from the equations and were left with a single equation in terms of \(x\).- Solving this equation provided us with possible \(x\) values.- These \(x\) values were then substituted back into the equation for \(y^2\) to find corresponding \(y\) values.This method is particularly useful for system equations where direct solving for one variable is complex. By substituting one into another, you effectively reduce the system's complexity, making calculations more manageable.