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Class 10 Maths notes for Linear Equation in two variables




What is Linear equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable. Linear equations can have one or more variables
Example
$ax+b=0$
a and b are constant
$ax+by+c=0$
a,b,c are constants
In Linear equation ,No variable is raised to a power greater than 1 or used as the denominator of a fraction.
So, $ax^2 +b=0$ is not a linear equation
Similary $ \frac {a}{x}+b=0$ is not a linear equation

Linear equations  are  straight lines when plotted on Cartesian plane

Linear equations Solutions


S.no
Type of equation
Mathematical representation
Solutions
1
Linear equation in one Variable
$ax+b=0 \; , \; a \neq 0 $
a and b are real number
One solution
2
Linear equation in two Variables
$ax+by+c=0 \; , \; a \neq 0 \; and \; b \neq 0$
a, b and c are real number
Infinite solution possible
3
Linear equation in three Variables
$ax+by+cz+d=0 \; , \; a \neq 0 \;, \; b \neq 0 \; and \; c \neq 0$
a, b, c, d  are real number
Infinite solution possible

Graphical Representation of Linear equation in one and two variables

  • Linear equation in two variables is represented by straight line the Cartesian plane.
  • Every point on the line is the solution of the equation.
  • Infact Linear equation in one variable can also be represented on Cartesian plane, it will be a straight line either parallel to x –axis or y –axis
    $x-2=0$ , (straight line parallel to y axis). It means ( 2,<any value on y axis ) will satisfy this line
    $y-2=0$, ( straight line parallel to x axis ). It means ( <any value on x-axis ),2 ) will satisfy this line

Steps to Draw the Given line on Cartesian plane| how to graph the linear equations

  1. Suppose the equation given is
    $ax+by+c=0 \; , \; a \neq 0 \; and \; b \neq 0$
  2. Find the value of y for x=0
    $y=\frac {-c}{b}$
    This point will lie on Y –axis. And the coordinates will be $(0,\frac{-c}{b})$
  3. Find the value of x for y=0
    $x=\frac {-c}{a}$
    This point will lie on X –axis. And the coordinates will be $(\frac {-c}{a}, 0)$
  4. Now we can draw the line joining these two points

Simultaneous pair of Linear equation:

A pair of Linear equation in two variables
$a_1x+b_1y+c_1=0$
$a_2x+b_2y+c_2=0$
Graphically it is represented by two straight lines on Cartesian plane.

Simultaneous pair of Linear equation
Condition
Graphical representation
Algebraic interpretation
$a_1x+b_1y+c_1=0$
$a_2x+b_2y+c_2=0$
Example
$x-4y+14=0$
$3x+2y-14=0$
$\frac{a_{1}}{a_{2}}\neq \frac{b_{1}}{b_{2}}$
Intersecting lines. The intersecting point coordinate is the only solution
Graphical representation of Simultaneous pair of Linear equation where two lines intersect at one point
  One unique solution only.
$a_1x+b_1y+c_1=0$
$a_2x+b_2y+c_2=0$
Example
$2x+4y=16$
$3x+6y=24$
$\frac{a_{1}}{a_{2}}= \frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$
Coincident lines. The any coordinate on the line is the solution.
Graphical representation of Simultaneous pair of Linear equation where two lines are coincident to each other
Infinite solution.
$a_1x+b_1y+c_1=0$
$a_2x+b_2y+c_2=0$
Example
$2x+4y=6$
$4x+8y=18$
$\frac{a_{1}}{a_{2}}= \frac{b_{1}}{b_{2}}\neq \frac{c_{1}}{c_{2}}$
Parallel Lines
alt=
No solution
The graphical solution can be obtained by drawing the lines on the Cartesian plane.

Algebraic Solution of system of Linear equation in two variables

S.no
Type of method
Working of method
1
Method of elimination by substitution
1) Suppose the equation are
$a_1x+b_1y+c_1=0$
$a_2x+b_2y+c_2=0$
2) Find the value of variable of either x or y in other variable term in first equation
3) Substitute the value of that variable in second equation
4) Now this is a linear equation in one variable. Find the value of the variable
5) Substitute this value in first equation  and get the second variable
2
Method of elimination by equating the coefficients
1) Suppose the equation are
$a_1x+b_1y+c_1=0$
$a_2x+b_2y+c_2=0$
2) Find the LCM of a1 and a2 .Let it k.
3) Multiple the first equation by the value k/a1
4) Multiple the first equation by the value k/a2
4) Subtract the equation obtained. This way one variable will be eliminated and we can solve to get the value of variable y
5) Substitute this value in first equation  and get the second variable
3
Cross Multiplication method
1) Suppose the equation are
$a_1x+b_1y+c_1=0$
$a_2x+b_2y+c_2=0$
2) This can be written as
Using Cross multipication method  to solve Linear Equation in Two Variable
3) This can be written as
Using Cross multipication method to solve Linear Equation in Two Variable
4) Value of x and y can be find using the
x => first and last expression
y=> second and last expression

How to solve the Linear equations in Two variables

  1. First frame the two linear equation in two variable
  2. We can solve them either using elimination , substitution or cross multiplication method as explained above
  3. We can also solve them in graphical manner. We need ton draw the lines on the graph using the technique given above and then finding the intersection
  4. Always validate the solutions with both the equations at the end

Simultaneous pair of Linear equation in Three Variable

Three Linear equation in three variables
$a_1x + b_1y+c_1z+d_1=0$
$a_2x + b_2y+c_2z+d_2=0$
$a_3x + b_3y+c_3z+d_3=0$

Steps to solve the Linear equations in Three variables

  1. Find the value of variable z in term of x and y in First equation
  2. Substitute the value of z in Second and third equation.
  3. Now the equation obtained from 2 and 3 are linear equation in two variables. Solve them with any algebraic method
  4. Substitute the value x and y in equation first and get the value of variable z
Solved Examples
1)Solve the linear equation
$3(x+3)=2(x+1)$
Solution
$3x+9=2x+2$
or x+7=0
or x=-7
2)Solve the Simultaneous linear pair of equations
$x+y=6$
$2x+y=12$
Solution
We will go with elimination method
Step 1 ) Choose one equation
$x+y=6$
$x=6-y$
Step 2) Substitute this value of x in second equation
$2x+y=12$
$2(6-y)+y=12$
$12-2y+y=12$
or y=0
Step 3) Substitute of value of y in any of these equation to find the value of x
$x+y=6$
x=6
So  x=6 and y=0  Satisfy both the equations
3) Solve the equations
$x+2y =10 \; , \; 4x+8y=40$
Solution
As per algebraic condition
Condition
Algebraic interpretation
$\frac{a_{1}}{a_{2}}\neq \frac{b_{1}}{b_{2}}$
One unique solution only.
$\frac{a_{1}}{a_{2}}= \frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$
Infinite solution.
$\frac{a_{1}}{a_{2}}= \frac{b_{1}}{b_{2}}\neq \frac{c_{1}}{c_{2}}$
No solution
So here infinite solutions are possible

Steps involved in solving a linear equation word problem

  1. Read the problem carefully and note what is given and what is required and what is given. If it is not clear in first time,read it again.
  2. Denote the unknown by the variables as x, y, …….
  3. Translate the problem to the language of mathematics or mathematical statements.
  4. Form the linear equation in two variable using the conditions given in the problems.
  5. Solve the equation for the unknown either by substitution, elimination or cross multiplication methods
  6. Verify to be sure whether the answer satisfies the conditions of the problem.
Here are some of the Tips for specific word problems type
Problem type
Steps to be followed
Problem based on Distance, speed and time.
We need to remember the Speed formula in these problem to derive the correct mathematical statement
Speed = Distance /Time
Or
Distance = Speed X time
Or
Time = Distance /Speed
Problem Based on moving upstream and downstream in the river
We need to generally calculate the speed of Boat in still water and Speed of the stream. Let’s us assume them as x and y, then
For downstream movement, speed of the boat would
= x+y
For upstream movement, speed of the boat would
=x-y
Problem based on Geometry and Mensuration
 We need to remember the Angle sum property for Triangle, quadrilaterals, cyclic quadrilateral, parallelogram
For Triangle, Sum of angles =1800
For quadrilateral, Sum of angles=3600
Similarly
In case of parallelogram which is a special case of quadrilateral, the opposite angles are equal
In case of cyclic quadrilaterals, the opposite angles are supplementary
Problem based on Numbers and digits.
A two-digit number (xy   y -in unit place and x in tenth place) can be expressed as   10x+y
If the number are interchanged, it will be expressed as $10y+x$
So you can formulate the mathematical expression in the unknown using the above formula
For three-digit number(xyz), the expression would $(100x+10y+z)$
Problem based on Work Rate
The standard equation that will be needed for these problems is
Portion of Job done in given time= (Work rate of the person) X (Time taken working)
Age Problems
It is good to take present age   of the person as x and y.
Then k year ago, there age would be (x-k) and (y-k) respectively
Similarly, k year after, their age would be (x+k) and (y+k) respectively
So you can formulate the mathematical expression based on this and then solve the problem
Fractions problem
For   Fraction problem, if the numerator is x and denominator is y, then the fraction is x/y.  We can use the above expression to formulate the mathematical expressions
Mixing solution problems
In this problem type, we will be looking at mixing solutions of different percentages to get a new percentage.  The solution will consist of a secondary liquid (acid or alcohol) mixed in with water.  Amount of secondary liquid will be given by
Amount of secondary liquid= (percentage of secondary liquid) X (Volume of the solution)
Commercial mathematics problems
Commercial mathematics problems include interest rate, cost price, selling price problem.
Selling price or Marked price = Cost price + Profit
So x be the selling price and y be the cost price
Profit = y-x
In case selling at loss, loss would be =x-y
Similarly, simple interest problem could be formulated
Simple interest = (% Rate of interest) X (number of year) X (Principle amount)
Example
1) Asha is five years older than Rita. Five years ago, Asha was twice as old as Rita was then. Find their present ages.
Solution
Let present age of Asha be x years and present age of Rita be y years
Therefore, x = y + 5
or x – y = 5 ...(1)
5 years ago, Asha was x – 5 years and Rita was (y – 5) years old.
Therefore, x – 5 = 2(y – 5)
or x – 2y = – 5 ...(2)
Solving (1) and (2), we get y = 10 and x = 15
2) The perimeter of a rectangular garden is 20 m. If the length is 4 m more than the breadth, find the length and breadth of the garden
Solution
Let the length of garden = x m
and width of garden = y m
Therefore x = y + 4 ...(1)
Also, perimeter is 20 m, therefore
2(x + y) = 20
or x + y = 10 ...(2)
Solving (1) and (2), we get x = 7, y = 3
Hence, length = 7 m and breadth = 3m

Quiz Time

Question 1 What is the Solution of linear equation
$x +9y=12$
$5x+45y-60=0$
A) Infinite
B) No solution
C) one solution
D) None of the above
Question 2 What is the solution of system of Linear equations
$x+3y=6$
$4x+12y=18$
A) Infinite
B) No solution
C) one solution
D) None of the above
Question 3 What is the solution of system of Linear equations
$x+3y=6$
$x+12y=33$
A) Infinite
B) No solution
C) x=0 and y=3
D) x=3, y=0
Question 4 what does x-2=0 represent in Cartesian plane
A) Straight line parallel to x-axis
B) Straight line parallel to y-axis
C) Straight line passing through origin
D) none of these
Question 5 Which of these points is not on the line
$ax+by+c=0$
A) $x=0,y=\frac {-c}{b}$
B) $x=-c/a,y=0$
C)x=0,y=0
D) None of these
Question 6 Solve the linear equation?
$x+1=2(x-1) +9$
A) -6
B) 6
C)5
D) 0

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Reference Books for class 10

Given below are the links of some of the reference books for class 10 math.

  1. Oswaal CBSE Question Bank Class 10 Hindi B, English Communication Science, Social Science & Maths (Set of 5 Books)
  2. Mathematics for Class 10 by R D Sharma
  3. Pearson IIT Foundation Maths Class 10
  4. Secondary School Mathematics for Class 10
  5. Xam Idea Complete Course Mathematics Class 10

You can use above books for extra knowledge and practicing different questions.


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Class 10 Maths Class 10 Science

Practice Question

Question 1 What is $1 - \sqrt {3}$ ?
A) Non terminating repeating
B) Non terminating non repeating
C) Terminating
D) None of the above
Question 2 The volume of the largest right circular cone that can be cut out from a cube of edge 4.2 cm is?
A) 19.4 cm3
B) 12 cm3
C) 78.6 cm3
D) 58.2 cm3
Question 3 The sum of the first three terms of an AP is 33. If the product of the first and the third term exceeds the second term by 29, the AP is ?
A) 2 ,21,11
B) 1,10,19
C) -1 ,8,17
D) 2 ,11,20