CBSE Grade 10 Maths Chapter 3 - Pair of Linear equations in Two Variables

Linear Equations.

An equation is a statement that two mathematical expressions having one or more variables are equal.

Equations in which the powers of all the variables involved are one are called linear equations.

The degree of a linear equation is always one.

General form of a Linear Equation in Two Variables

A Linear Equation is an equation of straight line.

Linear Equation in two variables is in the form of

ax + by + c = 0

where a, b and c are the real numbers (a≠0 and b≠0) and,

x and y are the two variables.

Here a and b are the coefficients and c is the constant of the equation.

Example:

(i) 3x + 4y + 4 = 0

(ii) 2/3 x + y = 0

Representing linear equations for a word problem

To represent a word problem as a linear equation

Identify unknown quantities and denote them by variables.
Represent the relationships between quantities in a mathematical form, replacing the unknowns with variables.

Geometrical Representation of a Linear Equation

Geometrically, a linear equation in two variables can be represented as a straight line.

Solution of a Linear Equation in 2 variables

The solution of a linear equation in two variables is a pair of values, one for x and the other for y, which makes the two sides of the equation equal.

Example:

If 2x + y = 4, then (0,4) is one of its solutions as it satisfies the equation.

A linear equation in two variables has infinitely many solutions.

General form of Pair of Linear Equations in 2 variables

Two Linear Equations having two same variables are known as the pair of Linear Equations in two variables.

A pair of linear equations in two variables can be represented as follows

a₁x + b₁y +c₁ = 0

a₂x + b₂y +c₂= 0

where a₁, a₂, b₁, b₂, c₁ and c₂ are real numbers, such that a₁² + b₁² ≠ 0, a₂² + b₂² ≠ 0.

The coefficients of x and y cannot be zero simultaneously for an equation.

Example of pair of linear equations:

(i) 3x + 4y + 6 = 0 and x + 2y + 3 = 0

(ii) 13x - 7y + 9 = 0 and -4x + 12y + 3 = 0

Note :

Every solution of the linear equation is a point on the line representing it.

Each solution (x, y) of a linear equation in two variables, ax + by + c = 0, corresponds to a point on the line representing the equation, and vice versa.

A pair of values of variables ‘x‘ and ‘y’ which satisfy both the equations in the given system of equations is said to be a solution of the simultaneous pair of linear equations.

Solution of simultaneous Pair of Linear Equations.

A pair of linear equations in two variables can be represented and solved, by
(i) Graphical method
(ii) Algebraic method

(i) Graphical method. The graph of a pair of linear equations in two variables ,presented by two lines is used to find the solution..
(ii) Algebraic methods. Following are the methods for finding the solutions(s) of a pair of linear equations:

Substitution Method
Elimination Method
Cross-Multiplication Method.
Determinant Method

Graphical Method of Finding Solution of a Pair of Linear Equations

Graphical Method of finding the solution to a pair of linear equations is as follows:

Plot both the equations (two straight lines)
Find the point of intersection of the lines.

The point of intersection is the solution.

Comparing the ratios of coefficients of a Linear Equation

Consider the pair of linear equations in two variables

a₁x + b₁y +c₁ = 0

a₂x + b₂y +c₂= 0

where a₁, a₂, b₁, b₂, c₁ and c₂ are real numbers, such that a₁² + b₁² ≠ 0, a₂² + b₂² ≠ 0.

Now compare the ratios of the coefficients of the pair of linear Equations to determine the nature of the solutions.

Nature of a Pair of Linear Equations in a plane

For a pair of straight lines on a plane, there are three possibilities

i) They intersect at exactly one point

If the two lines intersect each other at one particular point then that point will be the only solution of that pair of Linear Equations. It is said to be a consistent pair of equations.

ii) They are parallel

If the two lines are parallel then there will be no solution as the lines are not intersecting at any point. It is said to be an inconsistent pair of equations.

iii) They are coincident

If the two lines coincide with each other, then there will be infinite solutions as all the points on the line will be the solution for the pair of Linear Equations. It is said to be dependent or consistent pair of equations.

Algebraic Method of Finding Solution of Consistent Pair of Linear Equations

The solution of a pair of linear equations is of the form (x,y) which satisfies both the equations simultaneously. Solution for a consistent pair of linear equations can be found out using

i) Elimination method

ii) Substitution Method

iii) Cross-multiplication method

iv) Determinant method

1. Substitution method

If we have a pair of Linear Equations with two variables x and y, then we have to follow these steps to solve them with the substitution method-

Step 1: We have to choose any one equation and find the value of one variable in terms of other variable i.e. y in terms of x.

Step 2: Then substitute the calculated value of y in terms of x in the other equation.

Step 3: Now solve this Linear Equation in terms of x as it is in one variable only i.e. x.

Step 4: Substitute the calculate value of x in the given equations and find the value of y.

Example : Solve the following pair of linear equations by substitution method

y – 2x = 1

x + 2y = 12

Solution :

(i) Express one variable in terms of the other using one of the equations.

So y = 2x + 1.

(ii) Substitute for this variable (y) in the second equation to get a linear equation in one variable, x.

x + 2y = 12

x + 2 × (2x + 1) = 12

⇒ 5 x + 2 = 12

(iii) Solve the linear equation in one variable to find the value of that variable.

5 x + 2 = 12
⇒ x = 2

(iv) Substitute this value in one of the equations to get the value of the other variable.

y = 2x + 1

y = 2 × 2 + 1

⇒y = 5

So, (2,5) is the required solution of the pair of linear equations y – 2x = 1 and x + 2y = 12.

2. Elimination method

In this method, we solve the equations by eliminating any one of the variables.

Step 1: Multiply both the equations by such a number so that the coefficient of any one variable becomes equal.

Step 2: Now add or subtract the equations so that the one variable will get eliminated as the coefficients of one variable are same.

Step 3: Solve the equation in that leftover variable to find its value.

Step 4: Substitute the calculated value of variable in the given equations to find the value of the other variable.

Example : Solve the following pair of linear equations by elimination method

x + 2y = 8 and 2x – 3y = 2

Solution :

Step 1: Make the coefficients of any variable the same by multiplying the equations with constants. Multiplying the first equation by 2, we get,

2x + 4y = 16

Step 2: Add or subtract the equations to eliminate one variable, giving a single variable equation.
Subtract second equation from the previous equation

2x + 4y = 16
2x – 3y = 2
– + –
———————–
0(x) + 7y =14

Step 3: Solve for one variable and substitute this in any equation to get the other variable.

y = 2,

x = 8 – 2 y

⇒ x = 8 – 4

⇒ x = 4

(4, 2) is the solution.

3. Cross multiplication method

The arrows indicate the pairs to be multiplied. The product of the upward arrow pairs is to be subtracted from the product of the downward arrow pairs.

By using this diagram we have to write the equations as given in general form then find the value of x and y by putting the values in the above notations.

4. Determinant method

• There are several situations which can be mathematically represented by two equations that are not linear to start with. But we alter them so that they are reduced to a pair of linear equations.