This document discusses linear equations in two variables. It defines a linear equation as one where each term is either a constant or the product of a constant and two variables. Linear equations can be represented by an equation of the form ax + by + c = 0. The graph of a linear equation is a straight line. Examples are provided to demonstrate how to use systems of linear equations to solve real-world problems involving costs. Solutions involve setting up systems of equations and determining whether they have a unique solution, infinitely many solutions, or no solution.
APEX INSTITUTE has been established with sincere and positive resolve to do something rewarding for ENGG. / PRE-MEDICAL aspirants. For this the APEX INSTITUTE has been instituted to provide a relentlessly motivating and competitive atmosphere.
Linear Algebra may be defined as the form of algebra in which there is a study of different kinds of solutions which are related to linear equations. In order to explain the Linear Algebra, it is important to explain that the title consists of two different terms. The very first term which is important to be considered in the same, is Linear. Linear may be defined as something which is straight. Linear equations can be used for the calculation of the equation in a xy plane where the straight lines has been defined. In addition to this, linear equations can be used to define something which is straight in a three dimensional perspective. Another view of linear equations may be defined as flatness which recognizes the set of points which can be used for giving the description related to the equations which are in a very simple forms. These are the equations which involves the addition and multiplication.
Polynomials And Linear Equation of Two VariablesAnkur Patel
A complete description of polynomials and also various methods to solve the Linear equation of two variables by substitution, cross multiplication and elimination methods.
For polynomials it also contains the description of monomials, binomials etc.
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APEX INSTITUTE has been established with sincere and positive resolve to do something rewarding for ENGG. / PRE-MEDICAL aspirants. For this the APEX INSTITUTE has been instituted to provide a relentlessly motivating and competitive atmosphere.
Linear Algebra may be defined as the form of algebra in which there is a study of different kinds of solutions which are related to linear equations. In order to explain the Linear Algebra, it is important to explain that the title consists of two different terms. The very first term which is important to be considered in the same, is Linear. Linear may be defined as something which is straight. Linear equations can be used for the calculation of the equation in a xy plane where the straight lines has been defined. In addition to this, linear equations can be used to define something which is straight in a three dimensional perspective. Another view of linear equations may be defined as flatness which recognizes the set of points which can be used for giving the description related to the equations which are in a very simple forms. These are the equations which involves the addition and multiplication.
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A complete description of polynomials and also various methods to solve the Linear equation of two variables by substitution, cross multiplication and elimination methods.
For polynomials it also contains the description of monomials, binomials etc.
Biological screening of herbal drugs: Introduction and Need for
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Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
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2. Introduction
• A linear equation is an algebraic equation in which
each term is either a constant or the product of a
constant and two variables.
An equation of the form ax +by + c = 0 where a , b ,
and c are real numbers ,such that a and b are not both
zero, is called a linear equation in two variables.
A linear equation in two variable has infinitely many
solution.
The graph of every linear equation in two variables is
a straight line.
3. CONTINUED
X=0 is the equation of the y-axis and y=0 is the
equation of x-axis.
The graph of x=0 is a straight line parallel to y-axis.
The graph of y=0 is a straight line parallel to the x-
axis.
An equation of the type y=mx represents a line
passing through the origin.
Every point on the graph of a linear equation in two
variables is a solution of the linear equation .
4. How its obtain?
The solutions of a linear equation can be obtained by substituting
different values for x in the equation to find the corresponding
values of y.
The values of x and y are represented as an order pair. To plot the
graph of a linear equation, its solutions are found algebraically and
then the points are plotted on the graph.
Any linear equation of the form 'ax + by + c = 0' represents a
straight line on the graph. The points of the straight line make up
the collection of solutions of the equation.
5. Algorithm
Obtain the linear equation . Let the equation the equation be ax +
by + c=0.
Give any three values to x and calculate the corresponding values
of y to obtain solutions If possible ,choose integral values of x in
such a way that the corresponding values of y are also integers.
6. Linear Equation in real life
One of the realities of life is how so much of the world runs by
mathematical rules. As one of the tools of mathematics, linear
systems have multiple uses in the real world. Life is full of situations
when the output of a system doubles if the input doubles, and the
output cuts in half if the input does the same. That's what a linear
system is, and any linear system can be described with a linear
equation.
7. EXAMPLE ONE
1. 4 chairs and 3 tables cost 2100 and
5 chairs and 2 tables cost 1750.
Find the cost of a chair and a table
separately.
SOLUTION Let a chair be x and that
of a table be y. Then,and
,4x + 3y = 2100, 5x + 2y = 1750
This system of equations,
a1/a2≠b1/b2
2. So,hence has a unique solution ,
dependent and consistent
8. EXAMPLE 2
30 pens and 50 pencils together cost
100, while 60 pens and 100 pencils
together cost 200. Find the cost of a pen
and that of a pencil.
SOLUTION Let the cost of a pen be x
and that of a pencil be y. Then,
30x + 50y = 100, 60x + 100y = 200
This system of equations:
a1/a2=b1/b2=c1/c2 ;Has infinitly many
solutions,dependent and consistant
9. EXAMPLE 3
35 fans and 25 tube lights together cost
15000, while 70 fans and 50 tube lights
together cost 29000. Find the cost of a
pen and that of a pencil.
SOLUTION Let the cost of a fan be x and
that of a tube light be y. Then,
35x + 25y = 15000, 70x + 50y = 29000
This system of equations:
a1/a2=b1/b2≠c1/c2 ;Has no solution
,independent and inconsistant
