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Linear equations

linear equations in one variable

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EQUATIONS
INTRODUCTION:
An equation is a statement which gives the relationship between two or more
quantities.
For example
(1) 2 + 3 = 5
(2) 3 x 4 = 12
(3) 6 – 2 = 4
The equations given above pertain to known quantities. However, the quantities may
be known or unknown. When x + 4 = 10, here 4 and 10 are the known quantities and x is an
unknown quantity.
When you say “my present age is two years more than twice my brother’s age”, this
verbal statement can be converted into an algebraic equation in the following way.
Let your brother’s present age = y
Let your present age = x
Then y = 2x + 2
Here x and y are known as variables. For different values of x, y has different values
corresponding to the values of x.
The letters a, b, c, x, y, z etc. are usually used to denote the unknown quantities or
variables.
A variable or a combination of variables and known quantities is also known as an
algebraic expression.
For example 2x + 5, 6y – 10, 3z etc.
In 2x + 5, 2 is known as the coefficient of x.
Basic Operations:
The four basic operations of arithmetic are Addition, Subtraction, Multiplication and
Division are also applicable to Variables.
Simple Equations:If in an equation, the power of the variable / variables is exactly one then
it is called a simple equation.
Solving an equation in one variable:
There are several steps involved in solving an equation.
1. Always ensure that the variables are on one side of the equation and the known
quantities (numbers) are on the other side of the equation.
2. Simplify the variables and the numbers so that each side of the equation contains
only one term.
3. Divide both sides of the equation by the coefficient of the variable.
Solving two simultaneous equations:
When two equations, each in two variables, are given, they can be solved for the
values of the variables in three ways.
(a) Elimination by Cancellation.
(b) Elimination by Substitution.
(c) Adding two equations and Subtracting one equation from the other.
Nature of Solutions:
When a pair of equations is given, there are three possibilities for the solution;
(a) unique solution
(b) infinite solutions
(c) no solution
Let the pair of equations be a1x + b1y +c1 = 0 and a2x +b2y +c2 = 0, when a1, b1, a2 and
b2 are the coefficients of x and y terms while c1 and c2 are the known quantities.
(a) A pair of equations having a unique solution:
If a1 / a2 not equal to b1 / b2, then there will be a unique solution.
We have solved such equations in the previous examples of this chapter.
(b) A pair of equations having infinite solutions:
If a1 / a2 = b1 / b2 = c1 / c2, then the pair of equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0
will have infinite solutions.
Note: In fact this means that there are no two equations as such and one of the two equations
as such and one of the two equations is simply obtained by multiplying the other with a
constant.
(c). A pair of equations having no solutions at all.
If a1 / a2 = b1 / b2 not equal to c1 / c2, then the pair of equations a1x + b1y + c1 = 0 and
a2x + b2y + c2 = 0 will have no solutions.
Note: In other words, the two equations will contradict each other or be inconsistent with
each other.
CHAPTER – LINEAR EQUATIONS
Level – I : Questions 1 to 25
1. In a school there are 850 students. If the number of girls is 56 less
than the number of boys, what is the number of boys in the school?
(a) 451 (b) 450
(c) 453 (d) 620
2. Solve
2 5 1
5 2 15x x
 
(a)
61
2
 (b)
63
2

(c)
2
63
 (d)
2
61

3. In a cricket match Sachin scored 15 runs more than Vinod. If
together they had scored 245 runs, how many runs did each of them
score?
(a) 115, 130 (b) 120, 125
(c) 110, 135 (d) 105, 140
4. The length of a rectangle exceeds its breadth by 7m. If the perimeter
of the rectangle is 94m, what is the length and the breadth of the
rectangle?
(a) 28 m, 22 m (b) 25 m, 24 m
(c) 30 m, 15 m (d) 27 m, 20 m
5. The ratio between the two complementary angles is 2 : 3. What is the
value of each angle?
(a) 36˚, 54˚ (b) 18˚, 27˚
(c) 30˚, 60˚ (d) 14˚, 76˚
6. The sum of the digits of a two-digit number is 9. If the digits of the
number are interchanged the number increased by 63, what is the
original number?
(a) 18 (b) 36
(c) 72 (d) 27
7. For
2 5 5
3 2 3
x
x



the value of x is :
(a) 1 (b) -1
(c) 0 (d) -2
8. If five times a number is subtracted for 20 the resultant is three times
the same number added to 4, what is the number?
(a) 2 (b) 5
(c) 8 (d) 6
9. The length of a rectangular plot is
1
2
2
times its width. If the perimeter
of the plot is 70m. What is the length of plot?
(a) 25 m (b) 15 m
(c) 33 m (d) 16 m
10. X has twice as much money as Y. Y has thrice as much money as Z.
If X, Y and Z together have Rs.1,000, what is the amount with X?
(a) Rs.100 (b) Rs.200
(c) Rs.400 (d) Rs.600
11. A man is 36 years old and his son is one-fourth as old as him. In how
many years will the son be four-seventh as old as his father?
(a) 24 years (b) 27 years
(c) 32 years (d) 21 years
12. The difference between the ages of X and Y is 15 years. If X’s age is
four times the age of Y, what is X’s age?
(a) 3 years (b) 5 years
(c) 20 years (d) 12 years
13. A number when divided by another number gives quotient as 6 and
remainder as 1. If the first number is 36 more than the second
number, what are the numbers?
(a) 40, 9 (b) 38, 2
(c) 43, 7 (d) 47, 11
14. In a rectangle, the length is thrice of its breadth. If the perimeter of
the rectangle is 32 cm, what is the length of the rectangle?
(a) 4 cm (b) 9 cm
(c) 12 cm (d) 16 cm
15. A father’s age is 20 years more than that of his son. Five years later,
the father’s age will be thrice as that of his son. What is the present
age of the father?
(a) 20 years (b) 25 years
(c) 30 years (d) 35 years
16. A two digit number has 3 in its units place and the sum of its digits is
1
7
th of the number itself. The number is:
(a) 33 (b) 43
(c) 53 (d) 63
17. If
2
3
a
x

 , find the value of ‘a’ from the equation 3x – 2 = 2x + 4.
(a) 14 (b) 15
(c) 16 (d) 17
18. An obtuse angle of a parallelogram is twice its acute angle. Find the
measure of each angle of the parallelogram.
(a) 60˚, 60˚, 120˚, 120˚ (b) 55˚, 55˚, 110˚, 110˚
(c) 70˚, 140˚, 50˚, 100˚ (d) 65˚, 130˚, 60˚, 120˚
19. When 16 is subtracted from twice the number, the result is 20 less
than thrice of the same number. What is the number?
(a) 1 (b) 2
(c) 3 (d) 4
20. Sum of two positive integers is 45. The greater number is twice the
smaller number. What is the smaller integer?
(a) 20 (b) 15
(c) 30 (d) 18
21. If the sum of three consecutive whole numbers is 384, what are the
numbers?
(a) 126, 127, 128 (b) 128, 129, 130
(c) 127, 128, 129 (d) None of these
22. Mr. Shah is 30 years older than his son who is 5 years old. After how
many years would Mr. Shah become 3 times as old as his son then?
(a) 7 years (b) 7 years 6 months
(c) 10 years (d) 8 years
23. Sum of two positive integers is 62 and their difference is 12. Find the
integers.
(a) 26, 36 (b) 25, 37
(c) 47, 15 (d) 30, 32
24. When a number is added to its half, we get 117. Find the number.
(a) 80 (b) 76
(c) 78 (d) 82
25. Find the ratio between the two numbers whose sum and difference are
25 and 5, respectively.
(a) 2 : 5 (b) 3 : 2
(c) 3 : 2 (d) 2 : 1
Level – II : Questions 26 to 50
26. A man is thrice as old as his son. After 14 years, the man will be
twice as old as his son. Find their present ages.
(a) 42 years, 14 years (b) 40 years, 13 years
(c) 36 years, 12 years (d) 45 years, 15 years
27. Divide 300 into two parts so that half of one part may be less than the
other by 48.
(a) 164, 136 (b) 106, 194
(c) 168, 132 (d) 186, 114
28. In an isosceles triangle, the equal sides are 2 more than twice of the
third side. If the perimeter of the triangle is 19. The length of one of
the equal sides is :
(a) 3 cm (b) 5 cm
(c) 8 cm (d) 6 cm
29. P has Rs.570 and has Rs.350 with them. How much money should Q
receive from P so that Q has Rs.120 less than 3 times what is left with
P?
(a) Rs.330 (b) Rs.320

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Linear equations

  • 1. EQUATIONS INTRODUCTION: An equation is a statement which gives the relationship between two or more quantities. For example (1) 2 + 3 = 5 (2) 3 x 4 = 12 (3) 6 – 2 = 4 The equations given above pertain to known quantities. However, the quantities may be known or unknown. When x + 4 = 10, here 4 and 10 are the known quantities and x is an unknown quantity. When you say “my present age is two years more than twice my brother’s age”, this verbal statement can be converted into an algebraic equation in the following way. Let your brother’s present age = y Let your present age = x Then y = 2x + 2 Here x and y are known as variables. For different values of x, y has different values corresponding to the values of x. The letters a, b, c, x, y, z etc. are usually used to denote the unknown quantities or variables. A variable or a combination of variables and known quantities is also known as an algebraic expression. For example 2x + 5, 6y – 10, 3z etc. In 2x + 5, 2 is known as the coefficient of x. Basic Operations: The four basic operations of arithmetic are Addition, Subtraction, Multiplication and Division are also applicable to Variables. Simple Equations:If in an equation, the power of the variable / variables is exactly one then it is called a simple equation.
  • 2. Solving an equation in one variable: There are several steps involved in solving an equation. 1. Always ensure that the variables are on one side of the equation and the known quantities (numbers) are on the other side of the equation. 2. Simplify the variables and the numbers so that each side of the equation contains only one term. 3. Divide both sides of the equation by the coefficient of the variable. Solving two simultaneous equations: When two equations, each in two variables, are given, they can be solved for the values of the variables in three ways. (a) Elimination by Cancellation. (b) Elimination by Substitution. (c) Adding two equations and Subtracting one equation from the other. Nature of Solutions: When a pair of equations is given, there are three possibilities for the solution; (a) unique solution (b) infinite solutions (c) no solution Let the pair of equations be a1x + b1y +c1 = 0 and a2x +b2y +c2 = 0, when a1, b1, a2 and b2 are the coefficients of x and y terms while c1 and c2 are the known quantities. (a) A pair of equations having a unique solution: If a1 / a2 not equal to b1 / b2, then there will be a unique solution. We have solved such equations in the previous examples of this chapter. (b) A pair of equations having infinite solutions: If a1 / a2 = b1 / b2 = c1 / c2, then the pair of equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 will have infinite solutions. Note: In fact this means that there are no two equations as such and one of the two equations as such and one of the two equations is simply obtained by multiplying the other with a constant.
  • 3. (c). A pair of equations having no solutions at all. If a1 / a2 = b1 / b2 not equal to c1 / c2, then the pair of equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 will have no solutions. Note: In other words, the two equations will contradict each other or be inconsistent with each other. CHAPTER – LINEAR EQUATIONS Level – I : Questions 1 to 25 1. In a school there are 850 students. If the number of girls is 56 less than the number of boys, what is the number of boys in the school? (a) 451 (b) 450 (c) 453 (d) 620 2. Solve 2 5 1 5 2 15x x   (a) 61 2  (b) 63 2  (c) 2 63  (d) 2 61  3. In a cricket match Sachin scored 15 runs more than Vinod. If together they had scored 245 runs, how many runs did each of them score? (a) 115, 130 (b) 120, 125 (c) 110, 135 (d) 105, 140 4. The length of a rectangle exceeds its breadth by 7m. If the perimeter of the rectangle is 94m, what is the length and the breadth of the rectangle? (a) 28 m, 22 m (b) 25 m, 24 m (c) 30 m, 15 m (d) 27 m, 20 m 5. The ratio between the two complementary angles is 2 : 3. What is the value of each angle? (a) 36˚, 54˚ (b) 18˚, 27˚ (c) 30˚, 60˚ (d) 14˚, 76˚
  • 4. 6. The sum of the digits of a two-digit number is 9. If the digits of the number are interchanged the number increased by 63, what is the original number? (a) 18 (b) 36 (c) 72 (d) 27 7. For 2 5 5 3 2 3 x x    the value of x is : (a) 1 (b) -1 (c) 0 (d) -2 8. If five times a number is subtracted for 20 the resultant is three times the same number added to 4, what is the number? (a) 2 (b) 5 (c) 8 (d) 6 9. The length of a rectangular plot is 1 2 2 times its width. If the perimeter of the plot is 70m. What is the length of plot? (a) 25 m (b) 15 m (c) 33 m (d) 16 m 10. X has twice as much money as Y. Y has thrice as much money as Z. If X, Y and Z together have Rs.1,000, what is the amount with X? (a) Rs.100 (b) Rs.200 (c) Rs.400 (d) Rs.600 11. A man is 36 years old and his son is one-fourth as old as him. In how many years will the son be four-seventh as old as his father? (a) 24 years (b) 27 years (c) 32 years (d) 21 years 12. The difference between the ages of X and Y is 15 years. If X’s age is four times the age of Y, what is X’s age? (a) 3 years (b) 5 years (c) 20 years (d) 12 years 13. A number when divided by another number gives quotient as 6 and remainder as 1. If the first number is 36 more than the second number, what are the numbers? (a) 40, 9 (b) 38, 2
  • 5. (c) 43, 7 (d) 47, 11 14. In a rectangle, the length is thrice of its breadth. If the perimeter of the rectangle is 32 cm, what is the length of the rectangle? (a) 4 cm (b) 9 cm (c) 12 cm (d) 16 cm 15. A father’s age is 20 years more than that of his son. Five years later, the father’s age will be thrice as that of his son. What is the present age of the father? (a) 20 years (b) 25 years (c) 30 years (d) 35 years 16. A two digit number has 3 in its units place and the sum of its digits is 1 7 th of the number itself. The number is: (a) 33 (b) 43 (c) 53 (d) 63 17. If 2 3 a x   , find the value of ‘a’ from the equation 3x – 2 = 2x + 4. (a) 14 (b) 15 (c) 16 (d) 17 18. An obtuse angle of a parallelogram is twice its acute angle. Find the measure of each angle of the parallelogram. (a) 60˚, 60˚, 120˚, 120˚ (b) 55˚, 55˚, 110˚, 110˚ (c) 70˚, 140˚, 50˚, 100˚ (d) 65˚, 130˚, 60˚, 120˚ 19. When 16 is subtracted from twice the number, the result is 20 less than thrice of the same number. What is the number? (a) 1 (b) 2 (c) 3 (d) 4 20. Sum of two positive integers is 45. The greater number is twice the smaller number. What is the smaller integer? (a) 20 (b) 15 (c) 30 (d) 18 21. If the sum of three consecutive whole numbers is 384, what are the numbers? (a) 126, 127, 128 (b) 128, 129, 130 (c) 127, 128, 129 (d) None of these
  • 6. 22. Mr. Shah is 30 years older than his son who is 5 years old. After how many years would Mr. Shah become 3 times as old as his son then? (a) 7 years (b) 7 years 6 months (c) 10 years (d) 8 years 23. Sum of two positive integers is 62 and their difference is 12. Find the integers. (a) 26, 36 (b) 25, 37 (c) 47, 15 (d) 30, 32 24. When a number is added to its half, we get 117. Find the number. (a) 80 (b) 76 (c) 78 (d) 82 25. Find the ratio between the two numbers whose sum and difference are 25 and 5, respectively. (a) 2 : 5 (b) 3 : 2 (c) 3 : 2 (d) 2 : 1 Level – II : Questions 26 to 50 26. A man is thrice as old as his son. After 14 years, the man will be twice as old as his son. Find their present ages. (a) 42 years, 14 years (b) 40 years, 13 years (c) 36 years, 12 years (d) 45 years, 15 years 27. Divide 300 into two parts so that half of one part may be less than the other by 48. (a) 164, 136 (b) 106, 194 (c) 168, 132 (d) 186, 114 28. In an isosceles triangle, the equal sides are 2 more than twice of the third side. If the perimeter of the triangle is 19. The length of one of the equal sides is : (a) 3 cm (b) 5 cm (c) 8 cm (d) 6 cm 29. P has Rs.570 and has Rs.350 with them. How much money should Q receive from P so that Q has Rs.120 less than 3 times what is left with P? (a) Rs.330 (b) Rs.320
  • 7. (c) Rs.260 (d) Rs.310 30. The difference between a two digit number and the number obtained by interchanging its digits is 81. What is the difference between the digits of the number? (a) 7 (b) 8 (c) 9 (d) 6 31. 2 mangoes and 5 oranges together cost Rs.15 and 4 mangoes and 3 oranges together costs Rs.23. find the individual cost of a mango and an orange. (a) Rs.4, Rs.2 (b) Rs.3, Rs.2 (c) Rs.5, Re.1 (d) Re.1, Re.1 32. The sum of numerator and denominator of a certain fraction is 11. If 1 is added to the numerator, the value of fraction become 1 2 , the fraction is: (a) 3 8 (b) 2 9 (c) 1 10 (d) 6 5 33. A workman is paid Rs.15 for each day he is present and is fined Rs.3 for each day, he is absent. If he works for x days in a month and earns Rs.360. The value of x is: (a) 22 (b) 24 (c) 25 (d) 21 34. The perimeter of an isosceles triangle is 23 cm. The length of its congruent sides is 1 cm less than twice the length of its base. The length of each side is: (a) 3 cm, 10 cm, 10 cm (b) 5 cm, 9 cm, 9 cm (c) 8 cm, 8 cm, 7 cm (d) 4 cm, 9 cm, 10 cm 35. The sum of two numbers is 2490. If 6.5% of one number is equal to 8.5% of other, then one of the numbers is: (a) 1008 (b) 1079 (c) 1411 (d) 3718
  • 8. 36. At a fair, a “bull’s eye” was rewarded 20 paise and a penalty of 8 paise imposed for missing the “bull’s eye”. A boy tried 50 shots and received only 48 paise. How many “bull’s eye” did he hit? (a) 15 (b) 17 (c) 16 (d) 18 37. A father is 25 years older than his son. Five years before, he was six times his son’s age at that time. How old is the son? (a) 6 years (b) 8 years (c) 10 years (d) 12 years 38. A father gives his son Rs.3,000 for a tour. If he extends his tour by five days, he has to cut down his daily expenses by Rs.20. The tour will now last for: (a) 25 days (b) 30 days (c) 20 days (d) 36 days 39. The sum of three fractions is 11 2 24 . When the largest fraction is divided by the smallest fraction, the fraction thus obtained is 7 6 which is 1 3 more than the middle fraction. The three fractions are: (a) 1 2 3 , , 2 3 4 (b) 2 3 4 , , 3 4 5 (c) 3 4 5 , , 4 5 6 (d) 3 5 7 , , 4 6 8 40. If A was half as old as B ten years ago and he will be two thirds as old as B in eight years, what are the present ages of A and B? (a) 28 years, 46 years (b) 22 years, 40 years (c) 24 years, 42 years (d) 12 years, 30 years 41. A certain number of articles are purchased for Rs.1,200. If price per article is increased by Rs.5, then 20 less articles can be purchased. What is the original number of articles? (a) 60 (b) 40 (c) 50 (d) 90 42. A person purchased a certain number of eggs at four a rupee. He kept one fifth of them and sold the rest at three a rupee and in the process gained a rupee. How many eggs did the person buy?
  • 9. (a) 30 (b) 40 (c) 50 (d) 60 43. An elevator has a capacity of 12 adults or 20 children. How many adults can board the elevator with 15 children? (a) 6 (b) 5 (c) 4 (d) 3 44. Two years ago, a father’s age was three times the square of his son’s age. In three years time, his age will be four times as that of his son’s age. What are their present ages? (a) 24 years, 6 years (b) 27 years, 3 years (c) 25 years, 7 years (d) 29 years, 5 years 45. A sum of Rs.10 is divided among a number of persons. If the number is increased by 1 4 th, each will receive 5 paise less. The number of persons is: (a) 20 (b) 10 (c) 40 (d) 80 46. Tina’s sister is younger to her by 3 years and her brother is elder to her by 4 years. The sum of the ages of Tina, her sister and her brother is 73 years. What is Tina’s age? (a) 22 years (b) 24 years (c) 20 years (d) 25 years 47. A half-ticket issued by railway costs half the full fare but the reservation charge is same on half-ticket as on full-ticket. One reserved first class ticket for a journey between two stations costs Rs.362 and one full and one half – reserved first class ticket costs Rs.554. The reservation charge is: (a) Rs.18 (b) Rs.22 (c) Rs.38 (d) Rs.46 48. A purse contains Rs.130 in equal denominations of 50 paise, 10 paise and 5 paise. How many coins of each type are there? (a) 100 (b) 200 (c) 150 (d) 250 49. A father said to his son. “I was as old as you are at present, at the time of your birth. “If the father’s present age is 38 years, what was his son’s age five years ago?
  • 10. (a) 14 years old (b) 19 years old (c) 38 years old (d) 33 years old 50. Ankita purchased a certain number of articles at a rate of 5 for 50 paise. If she had purchased them at a rate of 11 for one rupee, she would have spent 50 paise less. How many articles did Ankita purchase? (a) 15 (b) 25 (c) 55 (d) 45 key Chapter Linear Equations 1 c 2 b 3 a 4 d 5 a 6 a 7 c 8 a 9 a 10 d 11 b 12 c 13 c 14 c 15 b 16 d 17 c 18 a 19 d 20 b 21 c 22 c 23 b 24 c 25 b 26 a 27 c 28 c 29 c 30 c 31 c 32 a 33 c 34 b 35 c 36 c 37 c 38 b 39 d 40 a 41 b 42 d 43 d 44 d 45 c 46 b 47 b 48 b 49 b 50 c