HOD mathematics of DAV public school khanna & owner of Vk Education academy
Revision Notes on chapter 3 Pair of Linear Equations in two variables class 10
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Pair of Linear Equations in two variables
chapter 3
Linear Equations
A Linear Equation is an equation of straight line. It is in the form of
ax + by + c = 0
where a, b and c are the real numbers (a≠0 and b≠0) and x and y are the two variables,
Here a and b are the coefficients and c is the constant of the equation.
Pair of Linear Equations
Two Linear Equations having two same variables are known as the pair of Linear Equations in two variables.
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
Graphical method of solution of a pair of Linear Equations
As we are showing two equations, there will be two lines on the graph.
1. If the two lines intersect each other at one
particular point then that point will be the only solution of that pair
of Linear Equations. It is said to be a consistent pair of equations.
2. If the two lines coincide with each other, then
there will be infinite solutions as all the points on the line will be
the solution for the pair of Linear Equations. It is said to be
dependent or consistent pair of equations.
3. If the two lines are parallel then there will be no solution as the lines are not intersecting at any point. It is said to be an inconsistent pair of equations.
Algebraic Methods of Solving a Pair of Linear Equations
1. Substitution method
If we have a pair of Linear Equations with two variables x and y,
then we have to follow these steps to solve them with the substitution
method-
Step 1: We have to choose any one equation and find the value of one variable in terms of other variable i.e. y in terms of x.
Step 2: Then substitute the calculated value of y in terms of x in the other equation.
Step 3: Now solve this Linear Equation in terms of x as it is in one variable only i.e. x.
Step 4: Substitute the calculate value of x in the given equations and find the value of y.
2. Elimination method
In this method, we solve the equations by eliminating any one of the variables.
Step 1: Multiply both the equations by such a number so that the coefficient of any one variable becomes equal.
Step 2: Now add or subtract the equations so that the one variable will get eliminated as the coefficients of one variable are same.
Step 3: Solve the equation in that leftover variable to find its value.
Step 4: Substitute the calculated value of variable in the given equations to find the value of the other variable.
3. Cross multiplication method
Given two equations in the form of
a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where
We write it in general form as
To apply cross multiplication we use this diagram
The arrows indicate the pairs to be multiplied. The product of the
upward arrow pairs is to be subtracted from the product of the downward
arrow pairs.
By using this diagram we have to write the equations as given in
general form then find the value of x and y by putting the values in the
above notations.
y Interpretation of the pair of equations
Equations Reducible to a Pair of Linear Equations in Two Variables
There are some pair of equations which are not linear but can be reduced to the linear form by substitutions.
Given equations
We can convert these type of equations in the form of ax + by + c = 0
Letand
Now after substitution the equation will be
am + bn = c
It can be easily solved by any of the method of solving Linear Equations.
by shikha kaushal
Compound Interest Definition Compound interest is the interest calculated on the principal and the interest accumulated over the previous period. It is different from the simple interest where interest is not added to the principal while calculating the interest during the next period. Compound interest finds its usage in most of the transactions in the banking and finance sectors and also in other areas as well. Some of its applications are: Increase or decrease in population. The growth of bacteria. Rise or depreciation in the value of an item. Compound Interest Formula The compound interest formula is given below: Compound Interest = Amount – Principal Where the amount is given by: Where, A= amount P= principal R= rate of interest n= number of times interest is compounded per year It is to be noted that the above formula is the general formula for the number of times the principal is compounded in an year. If the amount is compounded annually, the amount is given as-
Revision Notes on Circle The equation of a circle with its center at C(x 0 , y 0 ) and radius r is: (x – x 0 ) 2 + (y – y 0 ) 2 = r 2 If x 0 = y 0 = 0 (i.e. the centre of the circle is at origin) then equation of the circle reduce to x 2 + y 2 = r 2 . If r = 0 then the circle represents a point or a point circle. The equation x 2 + y 2 + 2gx + 2fy + c = 0 is the general equation of a circle with centre (–g, –f) and radius √(g 2 +f 2 -c). Equation of the circle with points P(x 1 , y 1 ) and Q(x 2 , y 2 ) as extremities of a diameter is (x – x 1 ) (x – x 2 ) + (y – y 1 )(y – y 2 ) = 0. For general circle, the equation of the chord is x 1 x + y 1 y + g(x 1 + x) + f(y 1 +y) + c = 0 For circle x 2 + y 2 = a 2 , the equation of the chord is x 1 x + y 1 y = a 2 The equation of the chord AB (A ≡ (R cos α, R sin α); B ≡ (R cos β, R sin β)) of the circle x 2 + y 2 = R 2 is given by x cos ((α + β )/2) + y sin ((α - β )/2) = a cos (
We can convert these type of equations in the form of ax + by + c = 0 Let
and
Now after substitution the equation will be am + bn = c It can be easily solved by any of the method of solving Linear Equations. by shikha kaushal
