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Linear equations are mathematical expressions that form straight lines when plotted on a graph. They are called linear because of the use of the term 'line' in their characteristics. In these equations, both variables are raised to the power of one. They usually take the form: \[ax + by = c \] where:

• \(a\), \(b\) are constants (or coefficients).
• \(c\) is the constant term.
• \(x\) and \(y\) are variables.

For example, in the equation \(x + 3y = 4\), \(x\) and \(y\) represent the variables, while the numbers 1 and 3 are their coefficients, and 4 is the constant term. By solving linear equations, we find the specific values for \(x\) and \(y\) that make the equation true. To solve a linear equation, isolate one of the variables. In the provided example, you solve for \(x\) in terms of \(y\) to get \(x = 4 - 3y\). This expresses \(x\) purely in terms of \(y\), making it easier to substitute into another equation if needed, such as in systems of equations.

The quadratic formula is a tool used to find the roots of quadratic equations, which are equations of the form: \[ax^2 + bx + c = 0 \] where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). Quadratic equations can have zero, one, or two solutions. The quadratic formula is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula provides the solutions for \(x\) by undoing the square in the equation. To use it, simply substitute the coefficients of your equation into the formula. The notation \(\pm\) means you will often get two solutions: one where you add the square root, and one where you subtract it. This dual solution property is what allows quadratic equations to have two distinct roots. In the solution to our system, after substituting \(x = 4 - 3y\) into the nonlinear equation, we arrive at the quadratic equation \(13y^2 - 28y + 14 = 0\). By applying the quadratic formula, you find the potential solutions for \(y\).

A discriminant is a part of the quadratic formula denoted as \(b^2 - 4ac\). It provides crucial information about the nature of the roots of a quadratic equation. The value of the discriminant tells us how many roots the quadratic equation will have and their nature:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If the discriminant is zero, there is exactly one real root, also called a repeated or double root.
• If the discriminant is negative, the quadratic equation has no real roots, but rather two complex (or imaginary) roots.

In solving the equation \(13y^2 - 28y + 14 = 0\), the discriminant is calculated as \(b^2 - 4ac = 28^2 - 4 \cdot 13 \cdot 14\). The result is \(56\), which is positive, indicating that our quadratic equation will have two distinct real roots. Once we know how many solutions there are, we can proceed to calculate them using the full quadratic formula. Understanding discriminants helps learners grasp why solutions to quadratic equations take the form they do, as well as foresee the nature of potential solutions.