# Simultaneous Linear Equations

- Equation ax +by +c = 0 where a and b are both not zero is called a linear equation in two variables x and y.
- Two linear equations in two variables form a system of simultaneous linear equations.
- We dealt with systems which have one ordered pair (x, y) as a solution.
- A wide variety of word problems involving two unknown quantities can be solved by assuming two unknowns as x and y and then writing linear equations corresponding to given conditions.

## Exercise

*Solve the following simultaneous linear equations (1-5):*

- x +y -5 = 0, 2x -y +2 = 0.
- 2x = 5y +4, 3x -2y = -16.
- x = -1, 2x +3y -5 = 0.
- x -y = 0, x +y = 0.
- x +3y = 5, 7x -8y = 6.

*Solve the following questions:*

- The sum of two numbers is 40 and their difference is 8. Find the numbers.
- The sum of two numbers is 2 and difference is 20. Find the numbers.
- The sum of two numbers is 2 and difference is 1. Find the numbers.
- The sum of two numbers is 43. If the larger is doubled and the smaller is tripled, the difference is 36. Find the numbers.
- Two numbers are in ratio 4:7. If thrice the larger is added to twice the smaller, we get 87. Find the numbers
- Two numbers are in ratio 2:3. If 20 is added to each, the ratio becomes 6:7. Find the numbers.
- The sum of the digits of a two digit number is 12. If the digits are reversed, the new number is 12 less than double the original number. Find the number.
- A number of two digits exceeds four times the sum of its digits by 6 and it is increased by 9 on reversing its digits. Find the number.
- Four erasers and six pens cost Rs 36 while three erasers and nine pens cost Rs 45.Find the cost of one eraser and one pen.
- The cost of 5 kg of sugar and 7 kg of rice is Rs 153 and the cost of 7 kg of sugar and 5 kg of rice is Rs 147. Find the total cost of 6 kg of sugar and 10 kg of rice.
- There are 38 coins in a collection of 20 paise coins and 25 paise coins. If the total value of the collection is Rs 8.50, find the number of 25 paise coins.
- The sum of ages of A and B is 40 years. Five years ago, twice the age of A added to three times of age of B was 70 years. Find their present ages.
- Four years ago, Sumitra was three times as old as her daughter. Six years from now the mother will be twice as old as her daughter. Find their present ages.
- The ages of Ankita and Anshul are in ratio 2:1. After 4 years, the ages will be in ratio 3:2. Find their present ages.
- A car travels in 3 hours the distance covered by a bus in 4 hours. In half an hour, the car covers 10 kms more than the bus. Find their speeds.
- Solve x -3y = 3x -1 = 2x -y.

[**Hint.**x -3y = 3x -1 and 3x -1 = 2x -y]. - A man has some coins of 25 paise and some coins of 50 paise. The total number of coins is Rs 28 and the total value is Rs 10. Find the number of 25 paise coins.
- The sum of the digits of a two digit number is 5. The digit obtained by increasing the digit in tens place by unity is one-eighth of the number. Find the number.
- Six years hence a mans age will be three times his sons age and three years ago he was nine times as old as his son. Find their present ages.

## Answers

**1.**x = 1, y = 4

**2.**x = -8, y = -4

**3.**x = -1, y = 7/3

**4.**x = 0, y = 0

**5.**x = 2, y = 1

**6.**24 and 16.

**7.**11, -9.

**8.**3/2, 1/2

**9.**33 and 10

**10.**12 and 21

**11.**10 and 15

**12.**48

**13.**34

**14.**Eraser Rs 3 and pen Rs 4

**15.**Rs 206

**16.**18

**17.**25 years, 15 years

**18.**34 years, 14 years

**19.**8 years, 4 years

**20.**Car 80 km/hr, bus 60 km/hr

**21.**x = 2, y = -1

**22.**16

**23.**32

**24.**30 years, 6 years