Chapter 4 – Linear Equations In One Variable

2

3 Solution : Correct answer is 1. In examples 5 and 6, state whether the given statements are true (T) or false (F).

The shifting of a number from one side of an equation to other is called

• (a)Transposition
• (b)Distributivity
• (c)Commutativity
• (d)Associativity 5x 2x

If −4 = , then the numerical value of 2x – 7 is 3 5 19 13 13

• (a)
• (b)−
• (c)0
• (d)13 19 19

The value of x for which the expressions 3x – 4 and 2x + 1 become equal is

• (a)–3
• (b)0
• (c)5
• (d)1

If a and b are positive integers, then the solution of the equation ax = b has to be always

• (a)positive
• (b)negative
• (c)one
• (d)zero

Linear equation in one variable has

• (a)only one variable with any power.
• (b)only one term with a variable.
• (c)only one variable with power 1.
• (d)only constant term. If a statement is a proportion, the cross-products of the terms are equal. a c If = , then ad = bc. b d To solve equations like 5t = –20, you still need to get the variable t, on its own. The variable has been multiplied by a number, 5 — so you can get the variable on its own by dividing both sides by that numbers. In this case, you need to divide by 5. 5t = –20 Divide both sides by the same number 5t ÷ 5 = –20 ÷ 5 by which the variable is multiplied. t = –20 ÷ 5 t = –4

Which of the following is a linear expression:

• (a)x2 + 1
• (b)y + y2
• (c)4
• (d)1 + z

A linear equation in one variable has

• (a)Only one solution
• (b)Two solutions
• (c)More than two solutions
• (d)No solution 1 2

Value of S in +S= 3 5 4 1

• (a)
• (b)
• (c)10
• (d)0 5 15 −4 3

y = − , then y = 3 4 2 2 2 2 3 4 3 4

• (a)−
• (b)−
• (c)
• (d)4 3 4 3

The digit in the tens place of a two digit number is 3 more than the digit in the units place. Let the digit at units place be b. Then the number is

• (a)11b + 30
• (b)10b + 30
• (c)11b + 3
• (d)10b + 3

Arpita’s present age is thrice of Shilpa. If Shilpa’s age three years ago was x. Then Arpita’s present age is (a) 3(x – 3) (b) 3x + 3 (c)3x – 9 (d) 3(x + 3) A one-step equation is one that can be solved in one step by either adding, subtracting, multiplying or dividing by one thing. There are four main methods. For example:

• (i)a + 3 = 4.2 — solve by subtracting 3 from both sides to get a = 1.2
• (ii)s – 7 = 12 — solve by adding 7 to both sides to get s = 19
• (iii)9m = 27 — solve by dividing both sides by 9 to get m = 3
• (iv)d ÷ 8 = 2 — solve by multiplying both sides by 8 to get d = 16 Before you can solve an equation, you must be able to spot what kind of equation you have.

The sum of three consecutive multiples of 7 is 357. Find the smallest multiple.

• (a)112
• (b)126
• (c)119
• (d)116 In questions 16 to 32, fill in the blanks to make each statement true.

In a linear equation, the _________ power of the variable appearing in the equation is one.

The solution of the equation 3x – 4 = 1 – 2 x is _________.

The solution of the equation 2y = 5y – is _________.

Any value of the variable which makes both sides of an equation equal is known as a _________ of the equation.

9x – _________ = –21 has the solution (–2)

Three consecutive numbers whose sum is 12 are _________, _________ and _________.

The share of A when Rs 25 are divided between A and B so that A gets Rs. 8 more than B is _________.

A term of an equation can be transposed to the other side by changing its _________.

On subtracting 8 from x, the result is 2. The value of x is _________.

+ 30 = 18 has the solution as _________.

When a number is divided by 8, the result is –3. The number is _________.

9 is subtracted from the product of p and 4, the result is 11. The value of p is _________. 2 3

If x − 2 = 5 − x , then x = _________. 5 5

After 18 years, Swarnim will be 4 times as old as he is now. His present age is _________.

Convert the statement Adding 15 to 4 times x is 39 into an equation _________. A family spent Rs. 52.00 for circus tickets. This cost included a Rs. 3.25 service fee for the order, with the cost of the circus tickets being Rs. 9.75 each. How many tickets did the family buy? Justify your answer. Understand the problem The answer is the number of tickets that family bought. List the important information– The service fee is Rs. 3.25 per order, the tickets cost Rs. 9.75 each, and the total cost is Rs. 52. Let t represent the number of tickets bought. Total cost = Tickets + Service Fee 52.00 = 9.75t + 3.25 Make a Plan Think: First the variable is multiplied by 9.75, and then 3.25 is added to the result. Work backward to solve the equation. Undo the operations in reverse order. First subtract 3.25 from both sides of the equation and then divide sides of the new equation by 9.75. Solve 52.00 = 9.75t + 3.25 –3.25 –3.25 Subtract 3.25 from both sides. 48.75 = 9.75t 48.75 9.75t = Divide both sides by 9.75. 9.75 9.75 5=t The family bought 5 tickets. Look Back You can use a table to decide whether your answer is reasonable. Five tickets is a reasonable answer. Tickets Cost of Tickets Service charge Total cost 1 Rs. 9.75 Rs. 3.25 Rs. 13.00 2 Rs. 19.50 Rs. 3.25 Rs. 22.75 3 Rs. 29.25 Rs. 3.25 Rs. 32.50 4 Rs. 39.00 Rs. 3.25 Rs. 42.25 5 Rs. 48.75 Rs. 3.25 Rs. 52.00 Sometimes, a two-step equation contains a term or an expression with a denominator. In these cases, it is often easier to first multiply both sides of the equation by the denominator in order to remove it, and then work to isolate the variable.

The denominator of a rational number is greater than the numerator by 10. If the numerator is increased by 1 the and denominator is decreased by 1, then expression for new denominator is _________.

The sum of two consecutive multiples of 10 is 210. The smaller multiple is _________. In questions 33 to 48, state whether the statements are true (T) or false (F).

3 years ago, the age of a boy was y years. His age 2 years ago was (y – 2) years.

Shikha’s present age is p years. Reemu’s present age is 4 times the present age of Shikha. After 5 years Reemu’s age will be 15p years.

In a 2 digit number, the units place digit is x. If the sum of digits be 9, then the number is (10x – 9).

Sum of the ages of Anju and her mother is 65 years. If Anju’s present age is y years then her mother’s age before 5 years is (60 – y) years.

The number of boys and girls in a class are in the ratio 5:4. If the number of boys is 9 more than the number of girls, then number of boys is 9.

A and B are together 90 years old. Five years ago A was thrice as old as B was. Hence, the ages of A and B five years back would be (x – 5) years and (85 – x) years respectively.

Two different equations can never have the same answer.

In the equation 3x – 3 = 9, transposing –3 to RHS, we get 3x = 9.

In the equation 2x = 4 – x, transposing –x to LHS, we get x = 4. 15 15

If − 7 x = 9, then –7x = 9 + 8 8 x 7 x 6

If +1 = , then = 3 15 3 15

If 6x = 18, then 18x = 54 x 11

If = 15, then x = 11 15

If x is an even number, then the next even number is 2(x + 1).

If the sum of two consecutive numbers is 93 and one of them is x, then the other number is 93 – x.

Two numbers differ by 40, when each number is increased by 8, the bigger becomes thrice the lesser number. If one number is x, then the other number is (40 – x). Solve the following: 3x − 8

=1 2x 5x

=2 2x − 1 2x − 3 1

= 4x + 5 3 8 5

= x x −1 5(1 − x ) + 3(1 + x )

=8 1 − 2x 0.2x + 5 2

= 3.5x − 3 5 y − (4 − 3y ) 1

= 2y − (3 + 4y ) 5 x x −1

= 5 6

0.4(3x –1) = 0.5x + 1

8x – 7 – 3x = 6x – 2x – 3

10x – 5 – 7x = 5x + 15 – 8

4t – 3 – (3t +1) = 5t – 4

5(x – 1) – 2(x + 8) = 0 x 1 1 1 1

− x − = ( x + 1) + 2 4 3 6 12 1 1 5

( x + 1) + ( x − 1) = ( x − 2) 2 3 12 x +1 x − 2

= 4 3 2x − 1 3x + 1

= 5 3

1 –(x – 2) – [(x – 3) – (x – 1)] = 0 x −2 x −1

3x − =4− 3 4 3t + 5 4t − 3

−1 = 4 5 2y − 3 3y − 5 3

− =y+ 4 2 4

0.25 (4x – 5) = 0.75x + 8 9 − 3y 8

= 1 − 9y 5 3x + 2 3

=− 2x − 3 4 5x + 1 1

=− 2x 3 3t − 2 2t + 3 7

+ =t + 3 2 6 m −1 m −2

m − =1− 2 3

4 (3p + 2) – 5(6p –1) = 2(p – 8) – 6(7p – 4)

3 (5x – 7) + 2(9x – 11) = 4(8x – 7) – 111

0.16 (5x – 2) = 0.4x + 7

Radha takes some flowers in a basket and visits three temples one by one. At each temple, she offers one half of the flowers from the basket. If she is left with 3 flowers at the end, find the number of flowers she had in the beginning.

Rs. 13500 are to be distributed among Salma, Kiran and Jenifer in such a way that Salma gets Rs. 1000 more than Kiran and Jenifer gets Rs. 500 more than Kiran. Find the money received by Jenifer.

The volume of water in a tank is twice of that in the other. If we draw out 25 litres from the first and add it to the other, the volumes of the water in each tank will be the same. Find the volume of water in each tank.

Anushka and Aarushi are friends. They have equal amount of money in their pockets. Anushka gave of her money to Aarushi as her birthday gift. Then Aarushi gave a party at a restaurant and cleared the bill by paying half of the total money with her. If the remaining money in Aarushi’s pocket is Rs.1600, find the sum gifted by Anushka.

Kaustubh had 60 flowers. He offered some flowers in a temple and found that the ratio of the number of remaining flowers to that of flowers in the beginning is 3:5. Find the number of flowers offered by him in the temple.

The sum of three consecutive even natural numbers is 48. Find the greatest of these numbers.

The sum of three consecutive odd natural numbers is 69. Find the prime number out of these numbers.

The sum of three consecutive numbers is 156. Find the number which is a multiple of 13 out of these numbers.

Find a number whose fifth part increased by 30 is equal to its fourth part decreased by 30.

Divide 54 into two parts such that one part is of the other.

Sum of the digits of a two-digit number is 11. The given number is less than the number obtained by interchanging the digits by 9. Find the number.

Two equal sides of a triangle are each 4m less than three times the third side. Find the dimensions of the triangle, if its perimeter is 55m.

After 12 years, Kanwar shall be 3 times as old as he was 4 years ago. Find his present age.

Anima left one-half of her property to her daughter, one-third to her son and donated the rest to an educational institute. If the donation was worth Rs. 1,00,000, how much money did Anima have?

If is subtracted from a number and the difference is multiplied by 4, the result is 5. What is the number?

The sum of four consecutive integers is 266. What are the integers?

Hamid has three boxes of different fruits. Box A weighs 2 kg more than Box B and Box C weighs 10 kg more than Box B. The total weight of the three boxes is 48 kg. How many kilograms (kg) does Box A weigh?

The perimeter of a rectangle is 240 cm. If its length is increased by 10% and its breadth is decreased by 20%, we get the same perimeter. Find the length and breadth of the rectangle.

The age of A is five years more than that of B. 5 years ago, the ratio of their ages was 3:2. Find their present ages.

If numerator is 2 less than denominator of a rational number and when 1 is subtracted from numerator and denominator both, the rational number in its simplest form is 1 . What is the rational number?

In a two digit number, digit in units place is twice the digit in tens place. If 27 is added to it, digits are reversed. Find the number.

A man was engaged as typist for the month of February in 2009. He was paid Rs. 500 per day but Rs. 100 per day were deducted for the days he remained absent. He received Rs. 9,100 as salary for the month. For how many days did he work?

A steamer goes downstream and covers the distance between two ports in 3 hours. It covers the same distance in 5 hours when it goes upstream. If the stream flows at 3 km/hr, then find what is the speed of the steamer upstream?

A lady went to a bank with Rs. 1,00,000. She asked the cashier to give her Rs. 500 and Rs. 1,000 currency notes in return. She got 175 currency notes in all. Find the number of each kind of currency notes.

There are 40 passengers in a bus, some with Rs. 3 tickets and remaining with Rs.10 tickets. The total collection from these passengers is Rs. 295. Find how many passengers have tickets worth Rs. 3?

Denominator of a number is 4 less than its numerator. If 6 is added to the numerator it becomes thrice the denominator. Find the fraction.

An employee works in a company on a contract of 30 days on the condition that he will receive Rs. 120 for each day he works and he will be fined Rs. 10 for each day he is absent. If he receives Rs. 2300 in all, for how many days did he remain absent?

Kusum buys some chocolates at the rate of Rs. 10 per chocolate. She also buys an equal number of candies at the rate of Rs. 5 per candy. She makes a 20% profit on chocolates and 8% profit on candies. At the end of the day, all chocolates and candies are sold out and her profit is Rs. 240. Find the number of chocolates purchased.

A steamer goes downstream and covers the distance between two ports in 5 hours while it covers the same distance upstream in 6 hours. If the speed of the stream is 1 km/hr, find the speed of the steamer in still water.

Distance between two places A and B is 210 km. Two cars start simultaneously from A and B in opposite direction and distance between them after 3 hours is 54 km. If speed of one car is less than that of other by 8 km/hr, find the speed of each.

A carpenter charged Rs. 2500 for making a bed. The cost of materials used is Rs. 1100 and the labour charges are Rs. 200/hr. For how many hours did the carpenter work?

For what value of x is the perimeter of shape 77 cm?

For what value of x is the perimeter of shape 186 cm?

On dividing Rs. 200 between A and B such that twice of A’s share is less than 3 times B’s share by 200, B’s share is?

Madhulika thought of a number, doubled it and added 20 to it. On dividing the resulting number by 25, she gets 4. What is the number? 1. Ranika wanted her friend Radhika’s mobile number. But Radhika played a trick. She gave her the number as 9XYZP1Q2R3 and told her to decode it with the help of following equations : (a) 16 – 35 = 7 – 8 6Y − 7 1 (b) = 3Y + 9 3 Z 2 − 9 −5 (c) = 5 + Z2 9 3 13 (d) P + P = 10 10 (e) 4(Q + 4) = 5(Q + 2) (f) 3(R+10) + 200 = 236 2. Determine the missing value in the puzzle below : 3. Game : Who will be the Lakhpati ??? Rohit and Saurabh are playing a game. The one who solves the following equations will be a winner. Find out if you were at their place would you have been be a winner. Till what money did you reach successfully? Rules of the game (a) You can only move to the next problem if the previous answer is correct. (b) Winning amount slab Question Number Amount Won 1 Rs. 1,000 2 Rs. 2,000 3 Rs. 3,000 4 Rs. 4,000 5 Rs. 10,000 6 Rs. 12,000 7 Rs. 14,000 8 Rs. 20,000 9 Rs. 40,000 10 Rs. 1,00,000 (c) Problems: x +1 3

• (i)= 2x + 7 8 1 2
• (ii)+ =2 ( x − 1) ( x + 1) 6x + 1 x −3
• (iii)+1 = 3 6
• (iv)3m = 7m – −6 (v) –x = ( x − 10) 7 3 (vi) 5x + = x − 14 2 2 x 8 (vii) +1 = 3 15 x 3x 5x (viii) + − =2 2 4 6 (ix) + 4 = 14 2 x x (x) x + x+ = 97 − 3 7 2 4. Work with a partner. Modelling the Equation Material Required : Glasses and containers, Equation met 5x – 6 = 4 – – + + – – = + – – + Add 6 positive containers to each side of the equal and then remove the zero pairs. 5x – 6 + 6 = 4 + 6 + + = + + + + Arrange the glasses and 5x = 10 containers into five equal + + groups. + + Each Cup is matched with 2 positive contents. So, x = 2. Model the following equations: (i) 3x – 3 = 12 (ii) 12x + 4 = 24 (iii) 7y + 14 = 7 5. Cross word puzzle. Solve the given crossword and then fill up the given boxes. Clues are given below for across as well as downward filling. Also, for across and down clues, clue number is written at the corner of the boxes. Answers of clues have to be filled up in their respective boxes. Down 1. Inverse of addition. 4. A symbolic form made up of constants, variables and operation (other than algebraic expressions). 5. If a term of an expression consists of a number multiplied by one or more variables, this number is the __________________ of the term. 6. Inverse of division. 7. Equations that have the same solution. 8. An _____________ in an equality which is true for all values of the variable in the equality. Across 2. A statement formed when an equal sign in placed between two expressions. 3. 2(x+5) = 2x+10. This is an example of what property. 9. Branch of mathematics concerned with operation by symbolic numbers. 10. A linear equation of the form Ax + By = C when A and B both are not zero is in the _____________________. 11. An expression is _________________ if it has no grouping symbols and all the like terms have been combined. UNIT-4 LINEAR EQUATIONS IN ONE VARIABLE 125 MATHEMATICS Rough Work UNIT-4 Rough Work LINEAR EQUATIONS IN ONE VARIABLE 127 MATHEMATICS Rough Work