NCERT Class 10 Mathematics Linear Equations CBSE Board Sample Problems (For CBSE, ICSE, IAS, NET, NRA 2022)

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Linear Equations

S. No Type of Equation Mathematical Representation Solutions
S. noType of equationMathematical

representation

Solutions
1Linear equation in one Variable

a and b are real

number

One solution
2Linear equation in two Variable

a, b and c are real

number

Infinite solution possible
3Linear equation in three Variable

a, b, c and 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.

In fact, Linear equation in one variable can also be represented on Cartesian plane; it will be a straight line parallel either to x-axis or y-axis

(Straight line parallel to y-axis) . It means (2, < any value on y axis) will satisfy this line

(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

2) Find the value of y for

This point will lie on Y-axis. And the coordinates will be

3) Find the value of x for

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

4) Now we can draw the line joining these two points

Simultaneous pair of Linear equation:

A pair of Linear equation in two variables

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

Graphically It is Represented by Two Straight Lines on Cartesian Plane
Simultaneous pair of Linear equationConditionGraphical representationAlgebraic interpretation

Example

Intersecting lines. The intersecting point coordinate is the only solution
Intersecting Lines
One unique solution only

Example

Coincident lines. The any coordinate on the line is the solution.
Coincident Lines
Infinite solution.

Example

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

Algebraic Solution of System of Linear Equation
S. noType of methodWorking of method
1Method of elimination by substitution1) Suppose the equation are

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

2Method of elimination by equating the coefficients1) Suppose the equation are

2) Find the LCM of and . Let it k.

3) Multiple the first equation by the value

4) Multiple the first equation by the value

5) Subtract the equation obtained. This way one variable will be eliminated and we can solve to get the value of variable y

6) Substitute this value in first equation and get the second variable

3Cross Multiplication method1) Suppose the equation are

2) This can be written as

3) This can be written as

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

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