sangemrameshkumar: linear equations

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.

Nature of Solutions:

When a pair of equations is given, there are three possibilities for the solution;

(a) unique solution

(b) infinite solutions

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.

Quadratic Equations:

A quadratic equation is an algebraic equation in which the highest power of the unknown is two and the other powers of the unknown are only whole numbers.

The most commonly used form of a quadratic equation is ax² +bx + c = 0, where a ≠ 0, a, b and c are real numbers.

The following are the examples of quadratic equations:

2x² + 3x + 1=0

x² -3x – 10 =0

2x² – 8 = 0

A simple equation like 2x+5 = 15 will have only one solution, (i.e., x = 5) whereas a quadratic equation like 2x² + 5 = 13, will have two solutions, it means that x will have two values (i.e., + 2 and -2) that satisfy the equation.

For example, the quadratic equation, 2x² + 3x + 1 = 0 will be satisfied for x = -½ and x = -1.

The equation, 2x²–8=0 will be satisfied for the values x=+2 or x =-2.

The values of x that satisfy the given equation are called the roots of the equation.

Solving a quadratic equation

A quadratic equation can be solved by

(a) Factorization and

(b) Formula

Note: While factorizing the expression, x term should be written as the sum or difference of the factors of the product of the x² term and the constant term. This facilitates the factorization of the expression.

(b) Solving the quadratic equation by using the formula.

Formula is generally used where factorization is not easy.For the equation, ax²+bx+c = 0 the roots are

Sum and product of the roots

The sum and the product of the roots can be found without finding the roots of a quadratic equation.

For a quadratic equation ax + bx + c = 0, the sum of the roots is –b/a and the product of the roots is c/a.

Nature of the roots:

For the quadratic equation, ax + bx + c = 0, the roots are

The nature of the roots depends upon the value of

b²- 4ac, which is known as the discriminant.

If b²- 4ac > 0, the roots are real numbers and unequal.

If b²- 4ac < 0, the roots are imaginary and unequal.

If b²- 4ac = 0, the roots are real and equal.

Also if b²- 4ac is a perfect square, then the roots are rational.

Constructing a quadratic equation

               When the roots of the equations are given as α and β the equation is (x- α) (x- β) = 0

This can be simplified as x² – x (α + β) + αβ = 0.

               Sometimes instead of giving the roots, directly the sum of the roots and the product of the roots are given, then the equation can be formed as above.

Quadratic Expression

            ax²+ bx + c is called a quadratic expression.

Maximum and minimum value of a quadratic expression

            The quadratic expression ax²+ bx + c  will have a maximum or minimum value of  4ac - b²/ 4a and it occurs at x = -b/2a.  The expression will have a maximum value when a < 0 and a minimum value when a > 0.

Special Equations:

If there is only one equation in two variables, the equation does not have a unique solution. If some conditions are imposed on these unknowns or variables, the equation may have a unique solution. Such type of equations are called special equations.

For example, the equation 2x + 3y = 13 has infinite solutions. If a condition is imposed that x and y are positive integers, x = 2, y = 3 and x = 5, y=1 are the two solutions of the equation.

If one more condition (that x < y) is imposed, then there is only one solution, x = 2, y = 3 satisfying the equation.

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

2. Solve

(a)

(b)

(c)

(d)

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

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

5. The ratio between the two complementary angles is 2 : 3. What is the value of each angle?

(a) 36˚, 54˚ (b) 18˚, 27˚

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

7. For

the value of x is :

(a) 1 (b) -1

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

9. The length of a rectangular plot is

times its width. If the perimeter of the plot is 70m. What is the length of plot?

(a) 25 m (b) 15 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

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

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

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

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

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

16. A two digit number has 3 in its units place and the sum of its digits is

th of the number itself. The number is:

(a) 33 (b) 43

17. If

, find the value of ‘a’ from the equation 3x – 2 = 2x + 4.

(a) 14 (b) 15

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˚

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

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

21. If the sum of three consecutive whole numbers is 384, what are the numbers?

(a) 126, 127, 128 (b) 128, 129, 130

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

23. Sum of two positive integers is 62 and their difference is 12. Find the integers.

(a) 26, 36 (b) 25, 37

24. When a number is added to its half, we get 117. Find the number.

(a) 80 (b) 76

25. Find the ratio between the two numbers whose sum and difference are 25 and 5, respectively.

(a) 2 : 5 (b) 3 : 2

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

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

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

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

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

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

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

, the fraction is:

(a)

(b)

(c)

(d)

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

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

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

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

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

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

39. The sum of three fractions is

. When the largest fraction is divided by the smallest fraction, the fraction thus obtained is

which is

more than the middle fraction. The three fractions are:

(a)

(b)

(c)

(d)

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

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

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?

(a) 30 (b) 40

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

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

45. A sum of Rs.10 is divided among a number of persons. If the number is increased by

, each will receive 5 paise less. The number of persons is:

(a) 20 (b) 10

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

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

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

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?

(a) 14 years old (b) 19 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

key

Chapter Linear Equations

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41	b	42	d	43	d	44	d	45	c	46	b	47	b	48	b	49	b	50	c

sangemrameshkumar

Thursday, October 9, 2014

linear equations

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