91Ó°ÊÓ

A system of equations involves finding values for the variables that satisfy all given equations simultaneously. In our case, we have two equations: \(x^2 - 2y = 1\) and \(x^2 + 5y = 29\). These equations contain two variables, \(x\) and \(y\), quadratic in nature. We aim to find specific pairs of \((x, y)\) that solve both equations at the same time.
To tackle this, we can employ the elimination method. This method involves aligning the equations in such a way that one variable can be eliminated by addition or subtraction, making the system less complex to solve. Here, we realized both equations have \(x^2\) terms that can be canceled out when subtracted from one another. By doing this, we simplify our system to focus solely on solving for \(y\). Once we know \(y\), we can plug it back into one of the original equations to find the corresponding \(x\) values.

Solving for variables in a system of equations usually involves a step-by-step process. In our problem, the elimination method is the first move to determine \(y\). By subtracting one equation from the other, achieving \(7y = 28\) allows us to quickly solve for \(y\) by dividing both sides by 7, giving \(y = 4\).
With \(y\) known, the next step is to plug this value back into either of the original equations to solve for \(x\). Substituting \(y = 4\) into \(x^2 - 2y = 1\) simplifies the equation to \(x^2 - 8 = 1\). Adjusting the constants gives \(x^2 = 9\). Taking the square root, we find \(x = 3\) or \(x = -3\).
This systematic approach ensures each variable is accurately determined, resulting in a comprehensive solution that satisfies both equations.

Quadratic equations play a major role in systems like ours, where variables are raised to the power of 2. A standard quadratic equation has the form \(ax^2 + bx + c = 0\) and can yield two solutions based on the properties of square roots.
After solving for \(y\), we encountered \(x^2 = 9\). In this specific quadratic scenario, solving for \(x\) requires taking the square root of both sides, leading to \(x = 3\) or \(x = -3\). These twin solutions emerge due to the inherent nature of quadratic equations where both positive and negative roots are valid.
Verifying these solutions within the context of the second equation in our system is essential. It ensures that both solutions are correct and consistent with all given conditions, confirming the integrity of the solution process. Remember, quadratic equations will often result in multiple solutions, a key aspect of why they can initially seem complex but produce rich results when solved methodically.