NCERT notes for class 10 Mathematics, chapter 3 - pair of linear equations will help you learn with fun. Below notes are comprised of concepts, cross-word exercises, problems and much more to boost your talent.

Every student needs to go through the below content before start their practice with the Linear equations chapter for class 10 at least two to three times. Notes are very helpful for the quick revision and have become an integral part of the studies.

1. Two linear equations in the same two variables are called a

Pair of linear equations is a pair of 2 equations consist two same variables.

General form is:

$a_1$ x + $b_1$y = $c_1$ and

$a_2$ x + $b_2$ y = $c_2$

Where $a_1, a_2, b_1,b_2,c_1$ and $c_2$ are real numbers where $a_1^2 + b_1^2$ $\neq$ 0 and $a_2^2 + b_2^2$ $\neq$ 0.

2. A set of 2 linear equations can be solved by below listed methods:

i) Graphical Method : The graph of each equation is a straight line.

There are few conditions to evaluate the solution set or we can say that solution for the given linear equation is based on below conditions:

Situation | Graph of equations | Solution Type |

Pair of linear equations is consistent | If lines intersect at a point | Unique solution |

Pair of equations is dependent (consistent) | If lines coincide | Infinitely many solutions |

Pair of equations is inconsistent | If lines are parallel | No solution |

ii) Algebraic Methods : One of the most popular methods to solve a pair of linear equations :

a) Substitution Method

b) Elimination Method

c) Cross-multiplication Method

Ali | Ali's Son | Expression | |

Present Age | x | y | |

7 years ago | x - 7 | y - 7 | x-7 = 7(y-7) |

3 years from now | x + 3 | y + 3 | x + 3 = 3(y + 3) |

(x - 7) = 7 (y – 7 )

x - 7 = 7y - 49

Combine like terms, we get

Combine like terms, we get

x - 7y = - 49 + 7

x - 7y = - 42 say equation (1)

Again, (x + 3) = 3 (y + 3)

x + 3 = 3y + 9

x - 3y = 9 - 3

x - 3y = 6 say equation (2)

**Step 3: Solve for x and y**

Use elimination method, to find the value of x and y

Subtract equation (2) from equation (1), we have

- 7y + 3y = -48

or -4y = - 48

or y = 48/4 = 12

Substitute y = 12 in equation (2),

x - 3 $\times$ 12 = 6

x - 36 = 6

or x = 42

Therefore, Ali and his son's present ages are 42 and 12 respectively.

**Question 2**: Sudheer's cricket coach told him to buy 3 balls and 6 balls for Rs. 3900 for first match. For second match again, he spent Rs 1300 to buy another bat and 3 more balls of the same kind. What is cost of one bat and one ball.**Solution**:**Step 1: Represent data in a table**

Let x be the cost of one bat and y be the cost of one ball.

**Step 2: Simplify expressions**

Now we have 2 algebraic expressions,

Subtract equation (2) from equation (1), we have

- 7y + 3y = -48

or -4y = - 48

or y = 48/4 = 12

Substitute y = 12 in equation (2),

x - 3 $\times$ 12 = 6

x - 36 = 6

or x = 42

Therefore, Ali and his son's present ages are 42 and 12 respectively.

Expression | ||

Cost of one bat | x | |

Cost of one ball | y | |

Cost of 3 bats and 6 balls | 3x + 6y | 3x + 6y = 3900 |

Cost of one bat and 2 balls | x + 2y | x + 2y = 1300 |

3x + 6y = 3900 ...equation (1)

x + 2y = 1300 ...equation (2)

**Step 3: Solve for x and y**

Use substitution method, to find the value of x and y

Isolate equation (2) for variable x

x = 1300 - 2y, substitute the value in the equation (1), we get

x + 2y = 1300 ...equation (2)

Isolate equation (2) for variable x

x = 1300 - 2y, substitute the value in the equation (1), we get

3 ( 1300 - 2y) + 6y = 3900

3900 - 6y + 6y = 3900

or 3900 = 3900

Hence, the system of given linear equations has infinity many solutions.

**Question 3**: Verify if below system is consistent

x + y = 5 and 2x + 2y = 10

**Solution**:

3900 - 6y + 6y = 3900

or 3900 = 3900

Hence, the system of given linear equations has infinity many solutions.

x + y = 5 and 2x + 2y = 10

Given equations are: x + y = 5; 2x + 2y = 10

To check if the system is consistent or not, first find the ratios of the coefficients of x , y and constant term, that is,

To check if the system is consistent or not, first find the ratios of the coefficients of x , y and constant term, that is,

$\frac{a_1}{a_2}$ = 1/2

$\frac{b_1}{b_2}$ = 1/2

$\frac{c_1}{c_2}$ = 5/10 = 1/2

From above result, we can conclude that, $\frac{a_1}{a_2}$ = $\frac{b_1}{b_2}$ = $\frac{c_1}{c_2}$ = $\frac{1}{2}$

Above system of equations has coincident pair of lines, which represent it has infinite solutions. Hence, given set of linear equations is consistent.

2x + y - 6 = 0, 4x - 2y - 4 = 0