NCERT Class 10 Mathematics Formula CBSE Board Sample Problems Part 4

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Linear Equations in Two Variables

An equation of the form ax + by + c = 0, where a, b and c are real numbers, such that a and b are not both zero, is called a linear equation in two variables

Important points to Note

Linear Equations in Two Variables

S.no

Points

1

A linear equation in two variable has infinite solutions

2

The graph of every linear equation in two variable is a straight line

3

x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis

4

The graph x=a is a line parallel to y -axis.

5

The graph y=b is a line parallel to x -axis

6

An equation of the type y = mx represents a line passing through the origin.

7

Every point on the graph of a linear equation in two variables is a solution of the linear equation. Moreover, every solution of the linear equation is a point on the graph

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

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

y second and last expression