Problem 25 - College Algebra with Corequisite Support

Question

For the following exercises, 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: Substitute Known Value

We are given two equations: the first is a nonlinear equation, \(x^2 - y^2 = 9\), and the second is \(x = 3\). Substitute \(x = 3\) into the first equation to find the value of \(y\).

Step 2: Calculate Expression

Replace \(x\) with 3 in the first equation, so it becomes \(3^2 - y^2 = 9\). This leads to \(9 - y^2 = 9\).

Step 3: Simplify the Equation

Subtract 9 from both sides, resulting in \(-y^2 = 0\). Simplifying gives \(y^2 = 0\).

Step 4: Solve for y

Take the square root of both sides to solve for \(y\). Since \(y^2 = 0\), we have \(y = 0\).

Step 5: Verify the Solution

Verify the solution by plugging \(x = 3\) and \(y = 0\) back into the original equations. Check if \(3^2 - 0^2 = 9\), which simplifies to \(9 = 9\). This confirms the solution is correct.

Key Concepts

Substitution Method

The substitution method is a powerful tool used to solve systems of equations, especially when one of the equations is straightforward to manipulate. To use the substitution method effectively, follow these basic steps:

• Identify an easy substitution: Find an equation that you can simplify easily or already has a variable expressed in terms of another.
• Substitute the expression: Replace the variable in the second equation with this expression. This helps reduce the number of variables in that equation, making it easier to solve.

In this exercise, the equation "\(x=3\)" immediately provides a value for \(x\). Simply substitute \(x=3\) into the nonlinear equation \(x^2-y^2=9\). This transforms the equation into "\(3^2-y^2=9\)", simplifying the problem significantly by reducing it to a single variable problem.

Simplifying Equations

Simplifying equations is an essential skill that helps in making complex problems more manageable. Once you've substituted a value, like in our step where \(x=3\) is plugged into \(x^2-y^2=9\), simplification follows.

• Perform operations: Do simple arithmetic operations, such as addition, subtraction, multiplication, or division, to condense the equation.
• Isolate variables: Work towards expressions that isolate variables on one side of the equation to directly solve for them.

In this exercise, substituting \(x=3\) yields an expression \(9-y^2=9\). Simplifying \(9-y^2=9\) further involves moving terms around to find \(y^2=0\). Simplification converts initially complex equations into straightforward solutions.

Verifying Solutions

Verification is the final and crucial step in solving any equation system. It confirms that the solutions satisfy the original equations.

• Re-substitute solutions: After obtaining potential solutions for variables, substitute them back into the original equations.
• Check all conditions: Ensure that all equations are satisfied with these solutions.

Here, we substitute \(x=3\) and \(y=0\) back into the original system: \(x^2-y^2=9\). Plugging these values, \(3^2-0^2=9\), verifies that the solution holds true, as \(9=9\). Successful verification confirms the correctness of your solution, ensuring you've solved the problem accurately.