Problem 25 - College Algebra

Question

Use any method to solve the nonlinear system. $$ \begin{array}{r} x^{2}-y^{2}=9 \\ x=3 \end{array} $$

Step-by-Step Solution

Answer

The solution is \(x = 3\) and \(y = 0\).

Step 1: Identify the Given Equations

We have two equations in the system. The first equation is \(x^2 - y^2 = 9\) and the second is \(x = 3\). Our task is to find values of \(x\) and \(y\) that satisfy both equations.

Step 2: Substitute x into the First Equation

From the second equation, we know \(x = 3\). Substitute \(x = 3\) into the first equation: \((3)^2 - y^2 = 9\).

Step 3: Simplify the Equation

Calculate \(3^2 = 9\). The equation becomes \(9 - y^2 = 9\). Simplify this further to get \(-y^2 = 0\).

Step 4: Solve for y

Rearrange the equation \(-y^2 = 0\) to find \(y\). Divide both sides by \(-1\): \(y^2 = 0\). The solution is \(y = 0\).

Key Concepts

Substitution Method

The substitution method is a widely used technique for solving systems of equations. It works by solving one equation for one variable and then substituting this value into another equation.
This process transforms the system into a single equation with one variable, which is easier to solve.
In our problem, the substitution method simplifies the nonlinear system by using the fact that **x = 3**. This value is substituted into the other equation, making the solution process straightforward.

• Start by isolating a variable (in this case, x) in one of the equations.
• Substitute the expression for this variable into the other equation(s).
• Solve the resulting equation for the remaining unknown(s).

By adopting this method, we effectively reduce the complexity of the original nonlinear system, leading us directly to a solution. This makes the substitution method a powerful tool in solving various equation systems.

Solving Equations

Solving equations involves finding the values of variables that make the equation true. In the context of systems of equations, the goal is to find a set of values that satisfies all given equations concurrently.
For nonlinear systems, this might involve higher-degree polynomials or more complex relationships between variables. In our problem, we faced quadratic terms like **x^2 - y^2 = 9**.
After substitution, our primary task was to solve the simplified equation for **y**. This involved rearranging the equation **-y^2 = 0** to find **y = 0**.

• Check each solution in the original equations to verify they indeed satisfy all parts of the system.
• Remember that nonlinear equations might have multiple solutions, but in this case, only **y = 0** works.

Solving equations is a critical skill in algebra, requiring practice to effectively manage both linear and nonlinear systems.

Algebraic Methods

Algebraic methods, such as substitution and elimination, are essential techniques for solving systems of equations. They can handle both linear and nonlinear systems by simplifying equations step-by-step.
In the exercise, we used substitution to eliminate one variable, turning a two-variable system into a single-variable equation.
This process simplifies problem complexity and makes it easier to find solutions.

• Substitution focuses on expressing one variable in terms of others to simplify the equation.
• Elimination, on the other hand, works by adding or subtracting equations to remove one variable.
• Both methods aim to reduce the problem to simpler forms, aiding in straightforward calculations.

Understanding and applying algebraic methods is crucial for progressing in algebra and tackling increasingly complex mathematical challenges.