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Class 10 Maths notes for Linear Equation



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 in a linear equation is raised to a power greater than 1 or used as the denominator of a fraction.

So  ax2 +b=0   is not a linear equation

Similary  (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≠0

a and b are real number

One solution

2

Linear equation in two Variable

ax+by+c=0 , a≠0 and b≠0

a, b and c are real number

 

Infinite solution possible

3

Linear equation in three Variable

ax+by+cz+d=0 , a≠0 ,b≠0 and c≠0

a, b, c, d  are real number

 

Infinite solution possible

 

 

Graphical Representation of Linear equation in one and two variable

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

1) Suppose the equation given is

ax+by+c=0 , a≠0 and b≠0

2) Find the value of y for x=0

y=-c/b

This point will lie on Y –axis. And the coordinates will be (0,-c/b)

3) Find the value of x for y=0

x=-c/a

This point will lie on X –axis. And the coordinates will be (-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

a1x+b1y+c1=0

a2x +b2y+c2=0

Graphically it is represented by two straight lines on Cartesian plane.

Simultaneous pair of Linear equation

Condition

Graphical representation

Algebraic interpretation

a1x+b1y+c1=0

a2x +b2y+c2=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

intersecting lines
 

One unique solution only.

a1x+b1y+c1=0

a2x +b2y+c2=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.

same lines

Infinite solution.

a1x+b1y+c1=0

a2x +b2y+c2=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

parallel lines

 

No solution

 

 

The graphical solution can be obtained by drawing the lines on the Cartesian plane.

Algebraic Solution of system of Linear equation

 

S.no

Type of method

Working of method

1

Method of elimination by substitution

1) Suppose the equation are

a1x+b1y+c1=0

a2x +b2y+c2=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

a1x+b1y+c1=0

a2x +b2y+c2=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

a1x+b1y+c1=0

a2x +b2y+c2=0

 

2) This can be written as

Cross multipication

 

3) This can be written as

Cross multipication

4) Value of x and y can be find using the

x => first and last expression

y=> second and last expression

 

 

 Simultaneous pair of Linear equation in Three Variable

  Three Linear equation in three variables

a1x + b1y+c1z+d1=0

a2x + b2y+c2z+d2=0

a3x + b3y+c3z+d3=0

 

Steps to solve the equations

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 Simultanous 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 algebric 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
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