Problem 46 - Algebra and Trigonometry

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 pairs of numbers that satisfy the given conditions are \((x, y)= (3,2), (-3,-2), (-3, 2), (3,-2)\).

Step 1: Define parameters and write the equations

From the exercise, we get two equations. 1). The difference between the squares of two numbers is \(5\). So, \(x^{2}-y^{2}=5\). 2). Twice the square of the second number subtracted from three times the square of the first number is \(19\). This translates to \(3x^{2}-2y^{2}=19\). The system of equations then becomes: \[\begin{cases}x^{2} - y^{2} = 5 \\3x^{2} - 2y^{2} = 19\end{cases}\]

Step 2: Isolate one of the variables

Let's isolate \(x^{2}\) in the first equation, \(x^{2}=y^{2}+5\).

Step 3: Substitute in the second equation

We substitute our isolated variable \(x^{2}\) from the first equation into the second equation. This gives us: \[3(y^{2}+5) - 2y^{2} = 19\].

Step 4: Solve the equation for y

Solving the equation for \(y\), we simplify the equation to: \(y^{2}+15-19=0\), which can be rewritten as \(y^{2}-4=0\). Solving for \(y\) we obtain two solutions: \(y=2, -2\).

Step 5: Substitute the y values in the first equation

Substitute each \(y\) value into \(x^{2}=y^{2}+5\) to solve for \(x\). For \(y=2\), we get \(x^{2}=2^{2}+5=9\), so \(x=3, -3\). In the other case, for \(y=-2\), we get \(x^{2}=(-2)^{2}+5=9\), so \(x=3, -3\). Therefore, the pairs of solutions to the problem are \((x, y)= (3,2), (-3,-2), (-3, 2), (3,-2)\).

Step 6: Verify the solution

The solutions can be verified by substituting them back into the original equations to ensure they satisfy both conditions.