NCERT Class 10 Mathematics Linear Equations CBSE Board Sample Problems

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Linear Equations

S.no Type of equation Mathematical representation Solutions

S.no

Type of equation

Mathematical

representation

Solutions

1

Linear equation in one Variable

a and b are real

number

One solution

2

Linear equation in two Variable

a, b and c are real

number

Infinite solution possible

3

Linear 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 equation

Condition

Graphical representation

Algebraic interpretation

Example

Intersecting lines. The intersecting point coordinate is the only solution

Intersecting lines.

Intersecting Lines

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One unique solution only

Example

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

Coincident lines

Coincident Lines

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Infinite solution.

Example

Parallel Lines

Parallel Lines

Parallel Lines

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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.no

Type of method

Working of method

1

Method of elimination by substitution

1) 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

2

Method of elimination by equating the coefficients

1) 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

3

Cross Multiplication method

1) 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

ysecond 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