91Ó°ÊÓ

The substitution method is a fundamental technique used to solve systems of equations, where one equation is manipulated to express one variable in terms of another.

In our exercise, we start with two equations: the quadratic equation \(3x^2 - 2y^2 - 1 = 0\) and a linear equation \(4x-y-3 = 0\). The goal is to find the values of \(x\) and \(y\) that satisfy both equations simultaneously.

Implementing the Substitution Method

Choosing the second equation to begin with, we isolate \(y\) to get \(y = 4x - 3\). This expression for \(y\) then replaces every instance of \(y\) in the first equation, which transforms the system into a single variable problem.

Once substituted, we focus on simplifying and solving the quadratic equation that emerges. It's important to check each step for errors in arithmetic to avoid propagating mistakes through the solution.

A quadratic equation is a polynomial equation of the second degree, typically written in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). These equations often have two solutions and they describe parabolic paths in graph form.

After substituting \(y\) in our example, we obtain a quadratic equation in \(x\). Simplifying it, we arrive at an appropriate form to apply methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula.

Solving Quadratic Equations

Factoring involves rewriting the quadratic as a product of binomials. Completing the square transforms the equation into a perfect square trinomial, offering a way to solve for \(x\) without factoring. If these methods are not suitable or difficult to apply, the quadratic formula is a reliable alternative that will always work as long as the equation's coefficients are real numbers.

The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), is derived from the process of completing the square on the general form of a quadratic equation and is used to find its roots. It's an essential tool when solving any quadratic equation that cannot be easily factored.

In our exercise, once we have simplified the quadratic equation, we might not be able to factor it. That's when we apply the quadratic formula; by substituting the coefficients from the equation into the formula, we solve for the possible values of \(x\). Remember, the symbol \(\pm\) indicates that there will be two solutions for \(x\), stemming from the positive and negative square roots.

Interpreting the Results

After determining the values of \(x\), we substitute these back into \(y = 4x - 3\) to find the corresponding values of \(y\). Since there could be two values for \(x\), we'll end up with two possible pairs of \((x, y)\) solutions for the system of equations. Verification is crucial; we plug these solutions back into the original equations to confirm their validity.