A Set of Ordered Pair is known as relation.
Linear InequalitiesBack to Top
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 VariablesBack to Top
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 FunctionBack to Top
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 VariableBack to Top
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 FunctionsBack to Top
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 EquationsBack to Top
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.