Friends! Today in this article we will discuss some important aspects of mathematics which are merely required for any 11th class students. The importance of 11th standard is raised because after this students have to go through board examinations which will actually decide the direction of their career. The major part of the grade XI math is covered by Algebra problems and linear equations are the must required term to solve various algebra queries or you can take help of algebra problem solver.
The 11th standard is the last stage where students finally execute the algebra problems and after that in grade XII, algebra works as a tool to solve other complex queries like matrices and determinants.
Let us first start with the role of linear equations in mathematics, as every student is aware of presentation and formation of linear equation but the point which makes this term essential while solving other complex problems is as “Every predefined formula or principle which students study and then apply to solve other queries, is made by doing the analysis of assumed principles on linear equation”. In earlier classes students learn various forms of mathematical equations which are in linear form that’s why their simplification gets easy because of direct implementation of standard formulas. But when students come into 11th standard, the queries include mathematical Algebraic Equations which are not in linear form. So the first thing which students need to do with those problems is their conversion from non-linear to linear form. This process is called Normalization. For implementing Normalization, students should know the actual standard form of that particular equation. Let us take an example of any mathematical equation and elaborate the whole concept for better understanding:
Suppose an equation is as: x2 + 2 = y2
The above equation is of 2-order whose standard linear form is as
Ax2 + By2 = C
So A= 1, B = 1 and C= 2 , equation already includes all the linear derivatives but it has to be written in arranged order as:
x2 + y2 = 2
If the given equation is as x2 + 2 = 0 then how will be the comparison done. In that case the equation is assumed to be as:
x2 +(0) y2 + 2 = 0
means here A = 1, B = 0 and C = 2
Students have to memorize all the predefined standard forms of mathematical equations because to implement graphing, the given problem should be in its standard form.
Now let us come to today’s main topic which is Graphing Linear functions, for performing the graphing process the first thing which need to be implemented is what we have talked above, the recognition of standard form of linear equation. When the problem is being converted into its linear form then the real work is started for graphing. Students need to find some essential terms which are as:
Intercepts, slope of the line, and a linear function.
A linear function is the presentation of any equation which includes the y intercept and slope of the line as:
y = mx + b
Here 'm' and 'b' are real integer coefficients which are being determined by the properties of the equation line and x, y are the numerical values which actually provides the required solution of the problem.
While moving through the graphing phenomenon we observe following essential points:
• The slope of the line will always equal m.
• The slope is defined as m = (y2 – y1)/(x2 – x1) for any two points on the line.
• The point (0, b) will always be the y-intercept.
Intercept of any line is the point at which the line of the equation or any function actually intersects the graphs. When we need to do graphing of any linear equation in 2D plane then only x and y intercepts are required. At the point of x intercept, the line intersects at x axis so all the y points are zero similarly when Y intercepts than all the x points of equation are zero.
The value of slope can be negative , so if it does then students can interpret the line in two ways:
• m = -(A/B) = (-A)/B which indicates that you move down A units and then right
B units.
• m = -(A/B) = A/(-B) it means that line move up A units first and then left B units.
Slope of the line is basically defined as the rise and run ratio, rise means how far a line moves from a point in y axis direction whereas run means how far line moves in x direction from one point to another.
Example: Calculate the slope of the line that passing through (4,-1) and (2, -2). Then use
this slope to help graph this line.
The slope is given by m = (y2 – y1)/(x2 – x1). So we substitute in our values to get
m = -2 – (-1)/ ( 2 – 4) = (-2 + 1)/ -2 = -1/ -2 = ½
Slope of the line is equal to the tangent of the line so if slope of the line is ½ then it can be written as:
tan a= ½ = m
a = tan -1 (1/2)
here 'a' is the angle at which the line is being aligned with x- axis.
This is how students can elaborate the slope of the line by using the endpoints of the line but now if we have value of slope of the line rather then endpoints of the line then for determining the equation of the line we use the following way:
The equation of a line that passes through (x1, y1) points and has slope m is given by:
m(x – x1) = y – y1
here (x, y) is any point which lies on the line and (x1, y1) is a specific point given already.
To explain the execution of this principle let us take another example:
Example: evaluate the equation of the line with slope m = 3 that passes through point (4, -1).
Then, write this equation in standard slope-intercept form.
First substitute m = 3 and x1= 4, y1= -1 into formula
m(x – x1) = y – y1 to get following
3(x – 4) = y - (-1)
This form is further simplifies as
3(x – 4) = y + 1
Now, to write this in slope-intercept form, we have to solve this equation for y.
and for this first distributive property is implemented as
3x – 12 = y + 1
Now, use the Addition Property of Equality to add (–1) to both sides to get
3x – 13 = y or y = 3x –13.
Note: You can quickly check your answer by verifying that the slope of this equation (m=3) matches the given slope (m=3) or not and also by substituting the given point (4,-1) into y = 3x - 13
to get
–1= (3 X 4 )– 13 and verify that it is a solution.
That’s how the required terms like intercepts and Slope of the line are evaluated. There is one more suitable way to perform graphing of any mathematical equation in a quick while, that is Graphing calculator. It is an Online math calculator which is available on various online math tutoring websites and only runs on Javascript supported Internet Browser. When students prefer the online math service to take the required help in solving math problems then online tutors provides some more online calculators to students by which they get the solution in short time in comparison of manual solving. The point which makes these tools more important is the optimized way which is used by online calculators to solve the complex math queries in quick time.
Graphing of mathematical equations is a vast topic in math because there are various kinds of math equations and each equation has different graphing function. So to learn more and everything in much detailed manner, students can rely on the most reliable platform for extra tuitions which is Online math Tutoring Websites.
Along with implementation of moderate ways for giving lessons these online educational services get success because of 24 x 7 hours availability. Students are also being satisfied of lesson sessions given by online tutors because these service providers have much strong and capable bench strength of highly skilled online math tutors categorized according to various grades tutors.
Everyone knows that students are not much comfortable with long scheduled learning lessons that’s why online tutoring services has managed short lessons in a continuous manner and students has the authority to review any lesson as many times they want. After explaining the fundamentals in lessons, tutor provides various worksheets to students and by solving related queries students mind get clear in respect of that particular topic.
These online tutoring services have that legal certification which makes it a trusted way to go through with. This is required because we all know that Internet is the platform to develop and execute moderate cause but because of this cyber crime is also growing.
In upcoming posts we will discuss about Syllabus of Grade XI and Understand Pythagorean triples. Visit our website for information on Gujarat secondary education board
The 11th standard is the last stage where students finally execute the algebra problems and after that in grade XII, algebra works as a tool to solve other complex queries like matrices and determinants.
Let us first start with the role of linear equations in mathematics, as every student is aware of presentation and formation of linear equation but the point which makes this term essential while solving other complex problems is as “Every predefined formula or principle which students study and then apply to solve other queries, is made by doing the analysis of assumed principles on linear equation”. In earlier classes students learn various forms of mathematical equations which are in linear form that’s why their simplification gets easy because of direct implementation of standard formulas. But when students come into 11th standard, the queries include mathematical Algebraic Equations which are not in linear form. So the first thing which students need to do with those problems is their conversion from non-linear to linear form. This process is called Normalization. For implementing Normalization, students should know the actual standard form of that particular equation. Let us take an example of any mathematical equation and elaborate the whole concept for better understanding:
Suppose an equation is as: x2 + 2 = y2
The above equation is of 2-order whose standard linear form is as
Ax2 + By2 = C
So A= 1, B = 1 and C= 2 , equation already includes all the linear derivatives but it has to be written in arranged order as:
x2 + y2 = 2
If the given equation is as x2 + 2 = 0 then how will be the comparison done. In that case the equation is assumed to be as:
x2 +(0) y2 + 2 = 0
means here A = 1, B = 0 and C = 2
Students have to memorize all the predefined standard forms of mathematical equations because to implement graphing, the given problem should be in its standard form.
Now let us come to today’s main topic which is Graphing Linear functions, for performing the graphing process the first thing which need to be implemented is what we have talked above, the recognition of standard form of linear equation. When the problem is being converted into its linear form then the real work is started for graphing. Students need to find some essential terms which are as:
Intercepts, slope of the line, and a linear function.
A linear function is the presentation of any equation which includes the y intercept and slope of the line as:
y = mx + b
Here 'm' and 'b' are real integer coefficients which are being determined by the properties of the equation line and x, y are the numerical values which actually provides the required solution of the problem.
While moving through the graphing phenomenon we observe following essential points:
• The slope of the line will always equal m.
• The slope is defined as m = (y2 – y1)/(x2 – x1) for any two points on the line.
• The point (0, b) will always be the y-intercept.
Intercept of any line is the point at which the line of the equation or any function actually intersects the graphs. When we need to do graphing of any linear equation in 2D plane then only x and y intercepts are required. At the point of x intercept, the line intersects at x axis so all the y points are zero similarly when Y intercepts than all the x points of equation are zero.
The value of slope can be negative , so if it does then students can interpret the line in two ways:
• m = -(A/B) = (-A)/B which indicates that you move down A units and then right
B units.
• m = -(A/B) = A/(-B) it means that line move up A units first and then left B units.
Slope of the line is basically defined as the rise and run ratio, rise means how far a line moves from a point in y axis direction whereas run means how far line moves in x direction from one point to another.
Example: Calculate the slope of the line that passing through (4,-1) and (2, -2). Then use
this slope to help graph this line.
The slope is given by m = (y2 – y1)/(x2 – x1). So we substitute in our values to get
m = -2 – (-1)/ ( 2 – 4) = (-2 + 1)/ -2 = -1/ -2 = ½
Slope of the line is equal to the tangent of the line so if slope of the line is ½ then it can be written as:
tan a= ½ = m
a = tan -1 (1/2)
here 'a' is the angle at which the line is being aligned with x- axis.
This is how students can elaborate the slope of the line by using the endpoints of the line but now if we have value of slope of the line rather then endpoints of the line then for determining the equation of the line we use the following way:
The equation of a line that passes through (x1, y1) points and has slope m is given by:
m(x – x1) = y – y1
here (x, y) is any point which lies on the line and (x1, y1) is a specific point given already.
To explain the execution of this principle let us take another example:
Example: evaluate the equation of the line with slope m = 3 that passes through point (4, -1).
Then, write this equation in standard slope-intercept form.
First substitute m = 3 and x1= 4, y1= -1 into formula
m(x – x1) = y – y1 to get following
3(x – 4) = y - (-1)
This form is further simplifies as
3(x – 4) = y + 1
Now, to write this in slope-intercept form, we have to solve this equation for y.
and for this first distributive property is implemented as
3x – 12 = y + 1
Now, use the Addition Property of Equality to add (–1) to both sides to get
3x – 13 = y or y = 3x –13.
Note: You can quickly check your answer by verifying that the slope of this equation (m=3) matches the given slope (m=3) or not and also by substituting the given point (4,-1) into y = 3x - 13
to get
–1= (3 X 4 )– 13 and verify that it is a solution.
That’s how the required terms like intercepts and Slope of the line are evaluated. There is one more suitable way to perform graphing of any mathematical equation in a quick while, that is Graphing calculator. It is an Online math calculator which is available on various online math tutoring websites and only runs on Javascript supported Internet Browser. When students prefer the online math service to take the required help in solving math problems then online tutors provides some more online calculators to students by which they get the solution in short time in comparison of manual solving. The point which makes these tools more important is the optimized way which is used by online calculators to solve the complex math queries in quick time.
Graphing of mathematical equations is a vast topic in math because there are various kinds of math equations and each equation has different graphing function. So to learn more and everything in much detailed manner, students can rely on the most reliable platform for extra tuitions which is Online math Tutoring Websites.
Along with implementation of moderate ways for giving lessons these online educational services get success because of 24 x 7 hours availability. Students are also being satisfied of lesson sessions given by online tutors because these service providers have much strong and capable bench strength of highly skilled online math tutors categorized according to various grades tutors.
Everyone knows that students are not much comfortable with long scheduled learning lessons that’s why online tutoring services has managed short lessons in a continuous manner and students has the authority to review any lesson as many times they want. After explaining the fundamentals in lessons, tutor provides various worksheets to students and by solving related queries students mind get clear in respect of that particular topic.
These online tutoring services have that legal certification which makes it a trusted way to go through with. This is required because we all know that Internet is the platform to develop and execute moderate cause but because of this cyber crime is also growing.
In upcoming posts we will discuss about Syllabus of Grade XI and Understand Pythagorean triples. Visit our website for information on Gujarat secondary education board
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