Problem 46 - College Algebra

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 \(5 .\) Twice the square of the second number subtracted from three times the square of the first number is \(19 .\) Find the numbers.

Step-by-Step Solution

Answer

The two sets of numbers are \((-2, 1)\) and \((3, -4)\).

Step 1: Formulate the System of Equations

From the problem statement, two equations can be formed as follows: \[x^2 - y^2 = 5\] and \[3x^2 - 2y^2 = 19\]. These are the equations that form the system to solve.

Step 2: Solve the System of Equations

Next, solve the system using substitution or elimination method. It's easier to use substitution in this case; express \(x^2\) from the first equation as \(x^2 = 5 + y^2\) and substitute in the second equation to get \[3(5 + y^2 ) - 2y^2 = 19\]. Solve this equation to find possible values for \(y\).

Step 3: Calculate the Solutions

Solving the equation formed in step 2, the possible values for \(y\) are \(1\) and \(-4\). Substituting these values into our first equation, the corresponding values for \(x\) are \(-2\) and \(3\) respectively.

Key Concepts

Quadratic Equations

Quadratic equations are a specific type of polynomial equation where the highest exponent of the variable is 2. These equations take the general form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants.
When solving systems of nonlinear equations, like in our problem, you often encounter quadratic equations. In the given exercise, the equations \(x^2 - y^2 = 5\) and \(3x^2 - 2y^2 = 19\) include terms with variables squared.
Understanding how to manipulate these equations through various algebraic methods is key to finding their solutions. Quadratic equations can be solved using methods such as factoring, completing the square, or using the quadratic formula. However, in systems of equations, substitution and elimination are often more effective.

Substitution Method

The substitution method is a powerful algebraic tool used to solve systems of equations. The goal is to express one variable in terms of the other from one of the equations, and then substitute this expression into the other equation(s).
In our problem, we first rearrange the equation \(x^2 - y^2 = 5\) to get \(x^2 = 5 + y^2\). This expresses \(x^2\) in terms of \(y^2\).
By substituting \(x^2\) with \(5 + y^2\) in the second equation \(3x^2 - 2y^2 = 19\), we simplify our system to one equation with one variable:

• \(3(5 + y^2) - 2y^2 = 19\)

This substitution transforms a complex system into a more straightforward equation that can be solved to find the value of \(y\). Once \(y\) is determined, it can be substituted back into \(x^2 = 5 + y^2\) to find \(x\).

Algebraic Problem Solving

Algebraic problem solving involves using various algebraic methods and techniques to find the solutions to equations and systems of equations. It requires a logical and systematic approach to resolving mathematical problems.
The exercise given exemplifies algebraic problem solving by employing the substitution method to simplify and solve a system of nonlinear equations. The order of operations, balancing equations, and consistently applying algebraic rules are key factors in this process.
One starts by identifying key information and relationships, like the difference between squares and expressions involving squares in our exercise.

• Form equations based on the relationships described
• Use substitution or elimination methods, as suitable, to simplify the system
• Solve the resulting simpler equations to find solutions for unknown variables

Critical thinking and patience are essential, as these methods require careful manipulation and verification of results.

Elimination Method

The elimination method is an alternative approach used for solving systems of equations. Unlike substitution, which involves expressing one variable in terms of another, elimination focuses on combining the equations in a way that removes one variable, allowing you to solve for the other.
For systems involving quadratic equations, elimination might not always be straightforward but can be highly effective when the equations are structured with similar expressions. In other contexts, the elimination method is applied by:

• Aligning like terms of the equations
• Multiplying one or both equations to match coefficients of the target variable
• Adding or subtracting equations to cancel out the target variable

While in our problem, the substitution method is more efficient, understanding elimination broadens problem-solving strategies, aiding in instances where substitution might not be convenient or feasible.