Hey! Friends, Previously we have discussed about ellipse area calculator and today we will discuss some relevant and important topics which are required for graphing of any mathematical equation. Graph is a diagram which exhibits the function of two or more sets of values by their respective co-ordinates and plotting of this diagram is called graphing. For graphing most importantly the slope at which line rises with respect to any axis is required. Slope means the steepness or inclination of the line at particular angle on the graph or in other words the ratio of rise of the line with run between two endpoints is called slope:
M = Rise/ Run
it is also equal to the ratio of difference of y points with difference of x points.
slope formula : M = (y2-y1)/ (x2-x1)
for example if two endpoints of the line are as A (1,2) and B( 2,4) then
M = (4-2)/ (2-1) = 2
let us now talk about Intercepts which are required to find the co-ordinates. Intercept of the line or any curve surface is the point at which it intersects or crosses the graph axis.
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Hello friends today we are going to discuss the topic How To Find The Equation Of A Line, for finding the equation of a line we need to have the knowledge of the three parameters but before talking about the parameters we will be seeing that what actually is the equation of line,
Y = mx +c
With the help of this equation we can the equation of line passing through two points with a slope, the direction of x will always be horizontal and the direction of y will always be vertical, in the above equation m is the slop of the given line, if a line passing through the origin then its equation will be
Y = mx
If the value of m is 3 then the equation of line is y – 3x = 0
Now suppose we are having two points as (x1, y1) and (x2, y2) and we are asked to find the equation of the line then we can find the equation of line as
y – y1= m (x – x1)
Here the value of m will be = y2 – y1 /x2 – x1
So we can rewrite our line equation as
y – y1 = (y2 – y1 /x2 – x1) * (x –x1)
if we have two points as (1,2) and (2,3) and we are asked two find the equation of line passing through these points then the equation will be
y – 2 = m (x – 1)
Now we have to find the value of m, so the value of m will be
m = y2 – y1 /x2 – x1
m = 3-2 /2-1
m = 1
Now we will put the value of m in above equation,
y -2 = x -1
y –x =1
This is the required equation of line.
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In mathematics, altitude is a geometry word. It is defined as the height of an object or point in relation to sea level or ground level. In geometry, it is defined as the length of the perpendicular line from a vertex to the opposite side of a figure. Altitude is also known as height. It is the measurement of someone or something from head to foot or from base to top. It is said to be the elevation above ground. It is the height of an object above the horizon. Altitude has several definitions.
Altitude is used to find the area of a triangle. Altitude of a triangle is defined as the perpendicular line or line which passes to the base of a triangle and must be at right angle to the opposite sides. Area of a triangle is defined as the total space occupied by the surface of a triangle and area of a triangle is given by:
= ½ x base x altitude
Here, altitude is the length of the perpendicular line from the base to the opposite side of a triangle.
There are two conditions which should satisfy to find the altitude of a triangle:
1. Altitude must starts from a vertex.
2. Altitude must be perpendicular to the base of a triangle or a side of a triangle.
We don’t have altitude in square and rectangle because altitude is one of the sides of the figure.
Altitude is also useful in Factoring Polynomials Calculator.
Calculate the area of a triangle if base = 5 inch and altitude = 10 inch?
Area of a triangle is given by:
=½ x base x altitude
=1/2 x 5 x 10 (inch)2
Last Altitude and Factoring Polynomials Calculator are also discussed in Tamilnadu Education Board.
A sphere is a round type of shape in three dimensions, it looks like a circle but it is very different from circle, like circle it also has radius and diameter. If we write the equation for circle then it would be x2 + y2 =a2. Here a is the radius of circle, as circle is a two dimensional shape so we have taken x and y and when we talk about sphere as we know well now that sphere is a three dimensional body so it will have three parameters x, y and z, so if we are asked to write the equation then it will be x2 + y2 +z2 =a2. Now as our topic is sphere volume for finding that we need to have good knowledge of radius of sphere, the formula for the radius of circle is given below,
V = 4/3π r3
Here π is a constant whose value will be 22/7 and 4/3 is also a constant , if we see here then volume of the sphere is totally depend on radius as the radius increases volume will increase and as the radius decreases it will decrease. If we see a problem in which we are asked to calculate the volume of a sphere and radius is given as seven so we can calculate the volume as
V = 4/3πr3
V = 4/3 *22/7 *7*7
Seven will cancel out with seven and we remain with
V = 4/3 *22*49
V = 1437.33
We can also derive one more formula if we will calculate the value of 4/3π, so the value of this will be 4.19 so new formula for the volume will be 4.19 r3. We can also calculate the volume by this formula as well
A line which is not prepared a curve is known as straight line.
As we know that the equation of straightline is generally in the form of:
Y = mx + b; or
Y = mx + c;
Where ‘m’ is the slope (gradient) of a line, slope and gradient both are same term.
And ‘b’ is the y- intercept.
Now we will see how to find the value of ‘m’ and ‘b’;
If we want to find the value of ‘b’ then we need to find where the line intersects the y- axis. Than we are find the value of ‘m’:
We know that ‘m’ is the slope of line, and then the value of ‘m’ is:
m = Change in y,
Change in x
Suppose change in ‘y’ is 5 and change in ‘x’ is 2 then we can find the value of ‘b’ and slope of a line;
We know that slope of line is:
m = Change in y,
Change in x
So put the value in the given formula:
m = 5
So the slope of the line is 5 and the value of b is 2.
So the equation of line by putting the value of slope and b is:
We know that the equation of line is:
Y = mx + b;
Put value of slope and y- intersect we get the equation of line.
Y = 5x + 2;
Now we will see slope of the straight line.
Suppose the coordinates of a straight line is p1, p2 and q1, q2 and ‘m’ is the slope of a line, and then we use following formula for finding the slope of a straight line.
m = q1 – q2,
p1 – p2
Suppose we have the values of coordinates is (3, 6) and (7, 9), then by using these coordinates find the slope of a line.
We know that the slope of a straight line is:
m = y1 – y2,
x1 – x2
Now put the values of these coordinates we get:
m = 6 – 9,
3 – 7
So the slope of a straight line is:
m = -3
So the slope of line is 0.74.
The perception of a slope is central to differential calculus. For non-linear functions, the small change in the rate varies along the curve. The slope or we can say its gradient of line (gradient means the points in the direction of the greatest rate with increase of the scalar field), which describe its steepness, incline or grade and the value which is higher indicates a steeper (that means having the sharp inclination), incline. It is not defined for horizontal or vertical lines.
In the slope of a line we have to follow some of the steps for finding the slope formula for the line.
Step1: – First we find the difference of both the ‘x’ and ‘y’ coordinates. And we have to place both the coordinates in the ratio.
Step 2: – After that we take two points on a line, because we are using two sets of order pair both the coordinates having values ‘x’ and ‘y’.
Step 3: – If a line has a negative slope then it goes down from left to right.
Step 4: – If a line has a positive slope then it goes up from left to right.
Step 5: – IF a line has is vertical then the slope is undefined.
The slope formula for given two points.
The two points are (p1, q1) and (p2, q2)
Then the slope of the line is denoted by ‘m’.
m = rise = change in ‘q’ = q2 – q1,
run change in ‘p’ p2 – p1,
Where, the value of (p1 ≠ p2),
In case of algebra, if ‘y’ is the linear function of ‘x’, then the coefficient of ‘x’ is known as slope of the line. If the equation of the line is given by
Y = mx + c; where m is the slope of the line. The line equation is known as slope – intercept form. Now we will see what is the Distance Formula?
The two points (x1, y1) and (x2, y2), the distance between these points is given by the formula:
Ã¢ÂÂ¨ d = √ (x2 – x1)2 + (y2 – y1)2. The ICSE class 12 sample papers is very useful for exam point of view.
In the previous post we have discussed about linear equations calculator and In today’s session we are going to discuss about slope worksheet, A slope describes the steepness, inclination of a line. In simple terms slope is defined as the ratio of rise when divided by the run between two points on the line. The slope worksheet help in finding the points of a line which in the plane consisting of the x and y axes which is represented by the letter ‘m’ and is defined as the change in y coordinate by the change in x coordinate between the two distinct point on the line. We can also write it as
m = —— = ——– .
here the symbol Ã¢ÂÂ (pronounced as delta) is used which means the change or the difference.
If the two points are given suppose (x1,y1) and (x2,y2) then the change in x from one to the another will be considered i.e. x2 – x1 (run) whereas the change in y will be y2 – y1 (rise) now the new formula takes place will be
y2 – y1
m = ———-
x2 – x1
these formula will not work in the case of vertical line.
To understand this in the more precise manner we will take one example:
suppose a line runs through two points : s = (2,1) and t(13,6) now as stated in the above formula we will follow the following steps.
Ã¢ÂÂy y2 – y1
m = —— = ———–
Ã¢ÂÂx x2 – x1
6 – 2
= ——– (substitute the value in the formula)
13 – 1
= 4/12 (calculation takes place)
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To understand about the linear equations calculator, which is the part of 7th grade math, we need to first understand how to get the solution of the linear equation. There exist a variable in the linear equation and to find the value of the given variable is called solving the linear equation. For this we simply need to separate the constant and the variables in the given linear equation so that the value of the variable is calculated. If we have a equation, which contains the variables and the constants on both sides of the equation, we will take the steps such that the all the terms containing the variables appears on the left side of the equation and the constants appear on the right side of the equation.
Let us make it more clearly with the help of the following example:
2x + 6 = 8x + 4
We will try to move all the variables on the one side of the equation and the constants on the other side of the equation. For this we will first subtract 2x from both sides of the equation and get :
2x + 6 – 2x = 8x + 4 -2x
We get :
6 = 8x – 2x + 4
6 = 6x + 4
Now we will subtract 4 from both sides of the equation to get :
6 – 4 = 6x + 4 – 4
Or we get :
2 = 6x
Now we divide both sides of the equation by 6 and get :
2 / 6 = 6x / 6
1/3 = x
Thus we get the solution to the given equation. If we put the value of x in the equation, lhs = rhs
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A line is defined by an equation as y = m x + c. Here in the given equation y is the point on the y – axis as well as x is the point on the x – axis and m is defined as a slope of the line or sometimes it is called as a tangent of the line. A tangent or slope of line occurs in the case when line is inclined at some angle and c is the intercept of the line that cuts the y – axis. In the line Equation we can define the slope as a Derivative. It means that slope of a line is described as a derivative of a function that is declared for a line like if a function of a line is defined as y = f ( x ) at a point x = a then the derivative of the given function is denoted as f ‘ ( a ) that is described as the slope of a tangent of the graph that has y = f ( x ). By selecting the point on the graph a user can easily plot the graph for derivative slope .
When a function f will be differentiated at the point x0 when the line is look as straight sometime it is called as tangent that is approximately near to the point x0 . The derivative of the slope of line at point x0 is denoted as f ‘( x0 ) or d f ( x 0 ) / d x .
f ‘( x0 ) = lim x→x0 f ( x ) – f ( x0 ) / x – x0
In the above expression f ( x ) – f ( x0 ) = Δ f and x – xo = Δ x
then the equation is look as f ‘( x0 ) = lim x→x0 Δ f / Δ x .
In the next session we are going to discuss Slope Formula.
In this session, we will discuss slope and Slope Formula. Slope of a line basically describes its inclination. A higher slope value shows a steeper incline. It’s a practical term and can not be defined perfectly for horizontal lines or vertical lines in theory.
Slope is normally explained by the ratio of the rise divided by the run between two points on a line.
Mathematically it can be defined as
“the slope of a line in a plane in 2-D having x and y axes is defined as the ratio of the δy and δx means change in the y coordinate divided by the corresponding change in the x axis between two points on a line”.
Let us see how to find Slope Formula:
The rise between two points is y2 – y1 = δy
And run is the difference between two points horizontally i.e. x2 – x1 = δx
The slope of the line generally denoted by m and given as
m = ( y2 – y1 ) / ( x2 – x1 ) ( here x2 ≠ x1)
m = δy / δx = rise / run
Let’s have a linear function
y = mx + b
Here m is the slope of the line. If the slope of the line is considered at the points ( x1, y1) and ( x2, y2 )
( y – y1) = m ( x – x1)
The slope of the line is explained by the linear equation
px + qy + r = 0
Where m = – p / q
Parallel lines always have a equal slope or if they are vertical and have undefined slope and normal lines always have negative reciprocal slope.
for non linear functions the rate of change of the function varies according to the curve. Mean to say the derivative of the function at a point is the slope of the line that is tangent to the curve.
dy / dx = limδx→0 ( δy / δx )
Slope of road is given by
m = 100 tan ( angle )
In the next session we are going to discuss Undefined Slope.
Slope of a line is defined as a measure of steepness. Sometimes a slope is defined as the tangent of the given line that is expressed as an equation y = m x +c where m is defined the slope of the line that is equal to the m = ( y / x ) – c where c is the intercept of the line and also tan m = perpendicular / base. Slope is also defined as m = ( change in value of y ) / ( change in value of x). It can be understood by an example y = – x + 2 where value of the slope is – 1 that represents the negative slope .
Negative slope of a line is described as when a line is goes down and from left to the right side and also the negative slope of a line is defined as x coordinate of a graph is increases and the y coordinate decreases means when these definitions are written in terms of expression they are occurred as value of m < 0 that means negative value of m .
If there are some given points of line as ( 1 , -1 ) and ( – 1 , 1 ) that are define the ( x1 , y1 ) and (x2 , y2) coordinates then calculation of a slope of line is as follows :
We have the formula of finding the slope m = y2 – y1 / x2 – x1
Here x1 = 1 , x2 = -1 , y1 = -1 and y2 = 1 , so by putting the values of x1 , x2 , y1 and y2 in the equation we get m = ( 1 ) – ( -1 ) / ( – 1 ) – ( 1 ) and by simplifying this equation the
m = 2 / -2 = – 1 and the slope of a given line that have the coordinate ( 1 , -1 ) and ( – 1 , 1 ) is -1. In the next session we are going to discuss Slope as a Derivative
Equations are the important part of mathematics, or we can say that they are the heart of mathematics. Equation can be defined as mathematical statements that are connected with equal to sign. Now, let’s talk about the topic i. e. Standard equation of line.
We deal with many equations like: – linear equations, polynomial equations etc. In this session we will discuss about Standard Equation of a Line that is a linear equation. Many types of linear equations exist; like:- linear equation in two variables, linear equation in graphically form, linear equation in two step etc. Here we will only talk about Standard Equation of a Line. Linear equation is a statement of equality which contains one or more unknown quantities or variables. In mathematics session if p and q are two real numbers such that y!0 then, we have learnt that an equation of the standard form p y + q z = 0 is called linear equation in one line. Where p and q are both are integers and y, z are variables. There are many types of properties of linear equation like: – addition, subtraction and multiplication etc.
We take some addition, subtraction examples to simplify equation of a line in standard form.
Example 1:- verify that x=2 is a root of the equation 5x-12 = -2
Solution: – substituting x=2 in the given equation, we get L.H.S =5*2-12 =10-12 = -2 =R.H.S
(x=2 is a solution of the equation 5x-12=-2).
Example 2:-solving equation a=2x+3 to solving equation of a line in standard form. Where x= 2.
Solution :- step 1:- a=2(2)+3
Step 2:- a =4+3
Step 3:- a=7
Example 2:- verify the equation of a line a = -2x+5 in standard form where x= 2.
Solution: – step 1 :- a =(-2*2) + 5 (we can add 2x are both side .2x+a=5 )
Step 2:- a=(-4) + 5
Step 3:- a= 1
Hence, a= 1 is a solution of the given system.
In the next session we are going to discuss Negative Slope
WHAT IS COORDINATE SYSTEM?
Coordinate system is used for determining a point in space. It uses 2 or more values to determine a point.
Coordinate system may be
1. Cartesian coordinate system(x,y)
2. Polar coordinate system(r,θ )
DEFINATION OF POLAR COORDINATES SYSTEM:
- A two-dimensional coordinate system is polar coordinate system
- In polar coordinate system, each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction.
POLE: The fixed point in coordinate system is called the pole.
POLAR AXIS: the ray from the pole in the fixed direction is the polar axis.
RADIUS: The distance from the pole is termed as the radial coordinate or radius
EQUATION OF A LINE
y= mx + b
m, b are constants . m is the slope and b is the intercept on y axis
Polar Equations of Lines
r = (m*rcos θ + b)/sin θ
where m is the slope of the line andb is the y-intercept. When θ = 0, the graph will be indeterminate.
Let us see how to convert the polar to Cartesian co-ordinate and vice versa.
It can be done by the following expressions:
sin (θ) = [y/r]
cos (θ) = [x/r]
x= r cos (θ)
y=r sin (θ)
Problem 1: Solve the polar coordinates (6, 90°) into Cartesian coordinates
Change the polar into Cartesian form
Here, r = 6 and θ = 90°
x = r cos (θ)
y = r sin (θ)
x = 6 cos 90°= 6 × 0
x = 0
y = 6 sin 90°
= 6x 1
y = 6
Cartesian form is (0, 6)
Uses of Polar Coordinates
1. used in astronomy for finding the circular and orbital motion of many things in universe.
2. used in navigation
In the next section, we are going to discuss standard equations of a line.
Hello students today we are going to discuss about the equation of a line passes through two points which is also known as two point form of a line. let us assume a line AB passes through two points A (x1,y1) and B ( x2,y2 ).then the Cartesian form of the equation can be represent as (x2-x1) * (y- y1)=(y2-y1) * (x-x1) it can also be written as (y- y1)=(y2-y1)/(x2-x1) * (x-x1) . Where (y2-y1)/(x2-x1) represent the slope of the line with the x-axis.now let us take some examples to understand the topic of the equation of a line from two points. Find the equation of a line passes through the point (2,3) and (-3,4) . as we know the equation of a line from two points can be written as (y- y1)=(y2-y1)/(x2-x1) * (x-x1) here we consider the point (2,3) as (x1,y1) and (-3,4)as (x2,y2). By putting the values in the two point form we get (y-4) = (4-3)/ (-3-2) * (x-2). Which can further written as (y-4) = -1/5 * (x-2). By cross multiplication the equation can be written as 5(y-4) = -1(x-2). Or we can write it as 5y +x =22. This is representing the equation of a line from two points. Let us take another two point form example by considering a line passes through the point (-1,-1) and (4,3). As we consider earlier let (x1,y1) = (-1,-1) and (x2,y2) = (4,3). By substituting the values in two point form we get (y+1) = (3+1)/ (4+1) * (x+1). Which can further written as (y+1) = 4/5 * (x+1). By cross multiplication the equation can be written as 5(y+1) = 4(x+1). Or we can write it as 5y -4x =-1. This is representing the equation of a line from two points. I hope with the help of the examples given a above the two point form of a line will become easier to understand. In The Next Topic We Are Going To Discuss Formula For Slope Of A Line And Finding The Slope Of A Line.
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In our today’s session we are going to discuss about the intercept form of the line. Before explaining the intercept form we first define intercept. Intercept is known as the point of intersection of the line on the axes (x-axis and y-axis). Mathematically the equation of a line which intercepts on x-axis at point (a, 0) and on y-axis at (0, b) can be represent as x/a +y/b = 1. This equation can be obtained by two point form also.
Let us solve Intercepts problem.
Find the equation of a line which intercept on x-axis at 5 and on y-axis at 3.
The line intercepts on x-axis at 5 and on y-axis at 3. By substituting the value of the intercepts on the intercept form as a=5 and b= 3. We get x/5 +y/3 =1. On further solving we will get 3x + 5y = 15.
Let us take another example on intercept form of the line. Convert the equation of the line 3x + 6y = 12 in intercept form.
For converting the equation of the line 3x + 6y = 12 in intercept form divide the equation by 12 so that we get 3x/12 +6y/12 = 1. It implies that x/4 +y/2= 1 . It is the perfect equation of the line in the intercepts form. The equation we get is intercepting on x-axis at (4,0) and y-axis at (0,2).
Let us solve another example: convert the equation 2(x+2) = 3(y+3) in intercept form. For solving we first simplify the equation so that we get 2x – 3y = 5. Further dividing the equation we get 2x/5 – 3y/5 =1. This is the intercept form of the line having intercept on x-axis at (5/2, 0) and y-axis at (0,-5/3). In The Next Session We Are Going To Discuss Polar Equations of Lines.
Hello friends, we are going to learn about the Finding the Slope of a Line. The slope of a line is defined in terms of ratio of changing in the y coordinate with respect to change in x. In simple terms a slope or gradient of a line is defined by its steepness or also called as inclination of a line. We can define it in terms of variable as if slope of a line describe as m and it have the line coordinate (x1 , y1) to ( x2 , y2) so with these coordinates Formula for Slope of a Line is:
m = y2 – y1 / x2 – x1 but it should be noted that ( x1 is not equal to x2 )
The slope of a line means how much angle it creates with respect to its horizontal base. It can be described as: m = tan ( Θ ) where Θ is a angle of inclination.
We can understand it by an example that if there is a line segment that has the coordinates ( 1 , -4 ) and ( -4 , 2 ) then the slope of this particular line is calculated as follows :-
We can label these coordinates as x1 = 1, y1 = – 4 , x3 = – 4 and y2 = 2
Then slope of the line segment m is calculated by using the formula defined above
slope = m = y2 – y1 / x2 – x1
m = 2 + 4 / – 4 – 1 = 6 / -5 = – 6 / 5
So the slope of the line m = – 6 / 5 which passes to the coordinates (1 , -4 ) and ( -4 , 2 ).
We can take another example in which we have the incline angle 45′ than we calculate the slope by using the formula slope = m = tan ( Θ ) = tan ( 45 ‘) = 1.
In the next topic we are going to discuss Intercepts.
Hello friends In the session today we will discuss about the slope intercept form of an equation.
Before solving some problems based on slope intercept form of a equation of a line we will define what the slope intercept form actually is. The slope intercept form of a equation is a form where the the given equation is solved for y in term of x. In short we can say that the slope intercept form of an equation can be written as y = mx+c
Where m is the slope of the line with the x axis and c is the intercept of the line on y axis.
This equation can also be written as y = m (x-d) where m also represents the slope of the line with x axis and d represents the intercept of the line on x axis.
The method of slope intercept form of a equation can be derived by the two points form of the line.
Since two points form of a line passes through the points (x1,y1), (x2,y2) can be written as
y-y1 =( y2-y1)/(x2-x1) *(x-x1)
As we know the slope of line passes through two point can be written as
m = ( y2-y1)/(x2-x1)
By putting the value of m in the above two points form of a line
we get y-y1 =m *(x-x1)
By further solving we get y = mx +c where c is the intercept on y axis
Let us try to understand the slope intercept with the help of some examples
y = -7x +5
This equation is in slope intercept form. The y intercept is (0,5) and the slope is -7.
Let us take another example
Rewrite the equation 3x-5y-10 = 0 in slope intercept form
Solution: the equation can be written as 5y = 3x-10
Or y = (3/5)*x- 2
So the equation has the slope 3/5 and intercept on y axis is (0,-2). This is a brief introduction about slope intercept form equation. In the next article we will discuss about point slope form.