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Graphing Linear Relations and Functions


A Set of Ordered Pair is known as relation.
Suppose we have (2, 5), (-3, 4), (12, -5), (5, 7), (0, 5), (6, 9), (1, 6);
The value lies on x – axis and it is known as Domain of given set and the value of y – axis is known as range of a set.
So in these values 2, -3, 12, 5, 0, 6, 1 are the domain of a set.
5, 4, -5, 7, 5, 9, 6 is range of a set.
Function is set of order pair in which one element of domain is exactly mapped with one element of range.
Now we see how we plot the graphs of linear Functions.
Step 1: First we take some function.
Step 2: Then we know that the values present in the domain are plotted towards the x – axis.
Step 3: And range value is plotted towards the y – axis.
Step 4: At last we join all the points in a graph.
Suppose we have (2, 3), (3, 4), (5, 6);
In these values 2, 3, 5 is the domain of the function. 3, 4, 6 is The Range of the function. So when we plot graph then the domain values are plotted towards the x – axis and range value towards the y – axis.
For Graphing linear Relations we follow the steps which are given below:
Step 1: In the relation the domain values are plotted towards the x – axis.
Step 2: The range value is plotted towards the y – axis.
Step 3: At last we join the points with a line and we get a graph of relation.
With the help of these steps we can easily plot the graph of relation and function.
This is the process of graphing linear functions.

Linear Inequalities

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Linear inequalities can be defined as expressions which include Combination of variables and constants as well as operators. Since it is an inequality it must contain either a greater than sign or a less than sign. These signs are represented as ‘<’ and ‘>’. It can also include less than equal and greater than equal, these are also included in inequalities. An inequality equation can be represented as x + 2y < 0. This, '<' symbol represents linear inequalities. Basically, inequality is a comparison between two Numbers or two expressions which can be determined and expressed in a different form. If two expressions are represented using equal symbol then it is said to be an equation which can be considered as an opposite of inequality. A linear Set of equations and their Transformations are considered to be a part of linear Algebra. It includes matrices, vectors, determinants etc. for solving any inequality equation, firstly we will consider that inequality as an equation and draw a graph for that equation then the inequality is placed and according to that region is shaded which appears in inequality.

Consider an equation x + 5 < 0. On taking 5 to right hand side of less than symbol, we get x < -5. On drawing graph of this inequality region below -5 will be shaded as solution of this inequality.
On representing linear inequality on number line, there are two types of intervals that are open interval and closed interval. In an open interval the end points or coordinates are not involved in interval while in a closed interval end points are considered in interval and considered to be a part of it. These points are represented by closed points or filled circles on number line.

Linear Equations with Two Variables

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Linear Equations with two variables are expressions in mathematics that consist of two unknown quantities in same equation and we need at least one more such equation to solve for unknowns. On substituting the values of variables in the equation we get the value of complete expression. These lines can be plotted on Cartesian plane by finding out the coordinate pairs of two variables. In these equations also we have the degree of equation as one i.e. it is a monomial.

To understand this concept of Linear Equations in 2 variables, let us consider a simple example of two equations given as follows: 52x + 44y = 100 and 26x + 72y = 20. To solve these linear equations we need to perform general algebraic operations. Multiplying second equation by 2 and subtracting it from second equation we get:

(52x + 44y = 100) – (52x + 144y = 40) = (- 100y = 80),
(- 100y = 80) is a linear equation in one variable. It can be for 'y' as shown below:
y = 80 / 100 (-1) = -4 / 5,
Substituting the value of 'y' in any of the two linear equation we get value of 'x'. Suppose we choose the equation 26x + 72y = 20, on substituting the value of 'y' in it we get:
26x + 72y = 20,
26 x + 72 (-4 /5) = 20,
26x = 20 + 288 / 5,
Or 26x = 388 / 5,
Or x = 194 / 65,
Thus solution of the two equations is x = 194 / 65 and y = -4/ 5. We can also say that two equations would intersect each other graphically at the Point (x, y) = (194/ 65, -4/5).

Slope in Linear Function

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Slope can be defined as a number that explains inclination of line which is going up or down. Slope is normally denoted by letter ‘m’. If line is going up that is if we are going from left to the right and during this movement, we go up then it will be said positive slope. If slope of line is going down that is we are going from right to left and coming down, it will be called as negative slope and if line is parallel to ground that is neither it is going up nor down then slope will be zero.

Let’s consider the following example to understand the concept of slope definition Math. Suppose slope is 2, means if we go 2 foot to right, the line goes up by 2 foot and shows the positive slope. If slope is -2, means if we go -2 units to right, the line goes down by 2 units and this shows the negative slope. Slope definition math can be understood by derivation of formula of slope.

For above figure, formula of slope (m) in slope math is given as:

Let’s consider two points Y (1) and Y (2) on Y – axis. Then we can write, d Y = Y (2) – Y (1). Similarly, consider two points X (1) and X (2) on X – axis then difference of these two points will be referred to run, and mathematically, d X = X (2) – X (1).

According to definition, slope of line is equals to Ratio of rise to run. Mathematically, we can define slope in math as shown below:
Slope (m) = d Y / d X
=Y (2) – Y (1) / X (2) – X (1),
If we consider angle ‘b’ then
Tan (b) = perpendicular / base = d Y / d X = Y (2) – Y (1) / X (2) – X (1).
When we compare two equations (both equations of ‘m’ and ‘tan (b)’). Hence m = tan (b).

Linear Equations with One Variable

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A system of Linear Equations can be defined as two or more linear equations which can be solved simultaneously. While solving linear equations Algebra 2 there are three possibilities of solutions:

1. one solution
2. no solution
3. infinite solution
Let’s have small introduction about solutions.

One solution: - If system given in two variable has one solution then it is an Ordered Pair and that ordered pair is a solution of both equations.

No solution: - If two lines 'x' and 'y' are parallel to each other then they will never intersect each other so system has no solution.

Infinite solution: - If two lines 'x' and 'y' lie on top of each other then we can say that there are infinite number of solutions.
If a linear system has two equations with two variables then system of linear equations can be written as: ap + bq = s and cp + dq = t, here any of the constant can be zero, it means each equation has at least one variable in it.

Now we will see how to solve linear system with one variables.

Suppose we have two equations with one variable i.e. 3p – 8 = 7 and 13 + 3q = 8.
Solution: Steps to follow to solve one variable equation are shown below.
Step 1: Take two equations that contain one variable in it. So we have 3p – 8 = 7 and 13 + 3q = 8.

Step 2: From both equations we can easily find the value of variable ‘p’ and ‘q’. So using 1st equation we can find value of 'p'. 3p – 8 = 7, so it can be written as:
3p = 7 + 8,
3p = 15
P = 15 / 3 = 3, we get value of 'p' as 3.

Step 3: Now take second equation, 13 + 3q = 8, it can be written as:
13 + 3q = 8,
3q = 8 + 13,
3q = 21,
q = 21 / 3,
q = 7, so here we get value of 'q' as 7.
This is how we solve one variable linear equation. This is all about Algebra 2 linear equations.

Special Functions

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Special Functions are mathematical functions that have one or more than one names. Special functions have many different names which are acceptable in mathematics. There is no specified definition of special functions; we can denote the special functions in many ways.

There is a list of functions that are accepted as special functions. Elementary functions are also considered as special functions.

When we solve differential equation then we get a function as result, we refer those functions as special functions.
Notations which are used in special functions are:

Functions like sine, cosine, tangent functions which are universally accepted are kind of special functions. So we can denote them by sin, cos and tan respectively, this is called special notation of the function. Sometimes a function has more than one name; we refer them as a special function. For Example: Function arctangent sometimes called as arc tan and sometimes as arctg, both names are special notation of function arctangent.

Sometimes superscript on trigonometric function denotes the exponent, but it can also modify the function. For Example:
Cos 3(x) can be written as (cos(x))3, both functions are same in functionality. We can write the function cos2(x) as (cos(x))2, both functions are same but we cannot write the function cos2(x) as cos (cos(x)). Just like this we can write the function cos-1 (x) as arccos(x) but we cannot write it as (cos(x))-1.
Special functions are functions with complex variables. Complex special function can be expressed in form of simple function.

Writing Linear Equations

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Linear equation is an algebraic function which consists of Cartesian coordinates x and y on plane. Linear equation with x and y coordinates gives a straight line when drawn on graph. We can draw the line on graph by placing the value of x and y coordinates on graph. Let us see standard form of writing a linear equation:
Steps for writing Linear Equations are:

1: Standard form of linear equation is:
y = mx + c.
x and y are coordinates of Straight Line, 'm' is the Slope of line and 'c' is y- intercept.

2: Now find the slope of line. We can calculate the slope of line using coordinates of line. Assume that starting points of line are (x1, y1) and ending points of line are (x2 , y2) then slope of line will be:
m = (y2 – y1) / (x2 – x1),
For Example: If coordinates are (3, 6) and (5, 8) then slope of line will be:
m = (8 - 6) / (5 – 3),
m = 1,

Step 3: Now we have to calculate the y- intercept of line i.e. 'c'. Y- Intercept is the Intersection Point with y- axis. For an example if intersection points are (0, 6) then y- intercept 'c' will be 6. This intercept can be either positive or negative. If intersecting point is above x- axis then interception point will be positive and if the intersection point is below x- axis then interception point will be negative.

Step 4: Now again write the equation y = mx + c. We put value of slope and y- intercept in this equation. This is how we write a linear equation.