12.2 Introduction to nonlinear equations

We have learned all about linear relationships, and linear functions. From what we learned about, we know that linear relationships and functions is consisted of two variables, one dependent, and the other independent. We also know that the line is a straight line. Linear equations are nonetheless the same, adding the criteria that its variables should not be raised to an exponent greater than one, and that the variables are not used as a denominator. We are to learn more of linear equations in this chapter, especially in section 1.

Apart from linear equations, we also need to learn about nonlinear equations. As the word suggests, these are the other equations that aren’t linear in nature, like the quadratic equations, circle equations, cubic equations, and more. We will have exercises in section 2 that will show us the difference between these equations and that of the linear equations.

As we have learned in the previous chapters, we know that linear equations show a straight line, that would appear to fall diagonally to on the Cartesian plane, to show the linear relationship between the values found in the line. However, in some special cases, linear equations can show horizontal lines and vertical lines. In chapter section 3 and 4 we will understand these special cases a bit more.

We are also going to learn how to find the parallel line and the perpendicular line of a given linear equation in section 5 and section 6. From our previous discussion in chapter, we learned that parallel lines have the same slope, whereas perpendicular lines have the slopes that are negative reciprocals with each other.

In section 7 and section 8 we have more practice in looking for the parallel lines and perpendicular lines, by looking for both of them for every equation given. There are also other applications of linear equation in section 8. To find out more about linear equations you can check out a video made by Education Alberta in Canada about “Exploring Linear Equations

Introduction to nonlinear equations

Non-linear equations, as it says in its name, are any functions that are not linear, for example, quadratic, circle and exponential functions. In this lesson, we will learn how to graph nonlinear equations, and then determine whether they are a function or not. The easiest way to verify if an equation is a function, no matter if it is linear or non-linear, is by using the vertical line test.


: Basically any function that is not "linear equation", such as quadratic, circle, reciprocal, exponential, etc.
    • a)
      y=x2+2y = x^2 +2
    • b)
      x=12y2x = {1 \over 2} y^2
    • c)
      y=2xy = {2 \over x}
    • d)
      y=x2y = \sqrt {x - 2}
    • e)
      y=x32y = x^3 - 2
    • f)
      x2y2=9x^2 - y^2 = 9
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Introduction to nonlinear equations

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