16801 equations

EQUATIONS

Quantitative Aptitude & Business Statistics

Quantitative Aptitude & Business Statistics: Equations

2

Introduction • Equation is defined to be

mathematical statement of equality . If the equality is true for certain value of variable involved ,the equation is often called a conditional equation and equality sign used (=);while if the equality is true for all values of the variable involved ,the equation called identity.


3

Simple Equations

• A simple equation is one unknown X is in the form aX+b=0

• Where and b are known constants .


4

Example

• Solve for X

• By transposing the variables in one side the constants in other side ,we have

519

15141

34

+=− xx

15

1915

143

4+=−

xx


5

Solution

12651524

524

156

5519

15)1420(

=××

=

=

+=

−

X

x

x


6

Example

• The denominator of a fraction exceeds the numerator by 5 and if 3 be added to both the fraction becomes ¾.Find the fraction


7

Solution

• Let X be the numerator and the Fraction be

• X/X+5 • By the Question

12243124

43

353

=+=+

=++

+

Xxxor

xx


8

• The required Fraction

1712


9

Example

• If thrice A’s age 6 years ago be subtracted from twice his present age, the result would be equal to his present age .Find A’s present age .


10

Solution

• Let x’ be years be A’s present age By the Question

2x-3(x-6)=x Or 2x-3x+18=x x=9 A’s present age is 9 years.


11

Example

• A number consists of two digits the digits in ten place is twice the digit in the units place. If 18 is subtracted from the number the digits are reversed. Find the number


12

Solution

• Let x be the digit in units place .So the digits in ten’s place is 2x.

• Thus the number becomes 10(2x)+x,By the question.

• 20x+x-18=10x+2x • 21x-18=12x • X=2


13

• So the required number is 10(2*2)+2=42


14

Simultaneous Linear Equations in two unknowns

• Methods of Solution • 1.Method of elimination • 2Cross Multiplication.


15

Elimination Method

• In this method two given linear equations are reduced to a linear equation in one unknown by eliminating one of the unknowns and solving for other unknowns.


16

Example

• Solve 2x+5y=9 and 3x-y=5

• x=2 • y=1


17

Cross Multiplication Method

• a1x+b1y+c=0 • a2x+b2y+c=0 x y 1 b1 c1 a1 b1 b2 c2 a2 b2

babaacacY

babacbcbX

babaacacY

cbcbX

221

1221

1221

1221

122112211221

;

)(1

)()(

−−

=−−

=

−=

−=

−


19

Example

• Solve 3X+2y+17=0 ;a1=3,b1=2.c1=17 5x-6y- 9 = 0 ; a2=5 b2=-6 c3=-9 X=-3,Y=-4


20

Quadratic Equation

• An equation of the form ax2+bx+c=0 where X is variable and a,b and c are constants with a≠0 is called a pure quadratic equation .

• The roots ;

aacbbX

242 −±−

=


21

Sum and product of roots

• Let the roots of equation be α and

• Sum of roots

• Products of roots

β

ab

−=+∞ β

ac

=∞β


22

How to construct the Quadratic Equation

• For the equation ax2+bx+c=0,we have known sum and product of roots

• X2-(Sum of roots )+product of roots=0


23

Nature of the Roots

• 1) If b2-4ac=0 the roots are real and equal

ii) If b2-4ac>0 the roots are real and unequal

iii) If b2-4ac<0 the roots are imaginary

aacbbX

242 −±−

=


24

• Note ;1- Irrational roots occur in pairs that is a root then is the other root of the equation

nm + nm −


25

Example

• Solve • X2-5x+6=0 a=1,b=-5 and c= 6

• X=3 and 2

aacbbX

242 −±−

=


26

Example

• If α and be the roots of X2+7x+12=0

• Find the equation whose roots are • and

β

2)( βα + 2)( βα −


27

Solution

• Hints • Sum of roots = 50 • Product of roots =49(49-48)=49 • Hence the required equation is X2-x(Sum of roots ) +product of roots

=0 X2-50x+49=0


28

Example

• If α and be the roots of • 2x2-4x-1=0 find the value of

β

αβ

βα 22

+


29

• Sum of roots =-b /a=-(-4)/2=2 • Product of roots =c/a=-1/2

223)( 33322

−=−+

=+

=+αβ

αββααβ

βαββ

αα


30

Example

• Solve

022.34 52 =+− +xx


31

Solution

0321203222.3)2(03222.3)2(

022.34

2

22

22

52

=+−

=+−

=+−

=+− +

yy

xx

xx

xx


32

Solution

• (y-8)(y-4)=0 • y=8 or y=4

• 2x=8 or 2x=4

• X=3 or 2


33

Example

• If one root of the equation is form the equation

• Solution • If one root is then the other

roots • Sum of roots =4 • Product of roots =1

32−

32−

32 +


34

Solution

• Required equation is • x2-x(Sum of roots ) +Product of

roots=0 • x2-4x+1=0


35

Example

• Divide 25 into two equal parts so that sum of their reciprocal is 1/6

0)10)(15(015025

61

2511

2

=−−=+−

=−

+

xxxx

xx


36

• X=10,15 • So the parts of 25 are 10 and 15


37

Solution of Cubic Equation

• Solve x3-7x+6=0 • Solution: • Putting X= 1 LHS is Zero So (x-1) is

a factor ,the other factors are finding by dividing this factor by LHS we get the expression (x2+x-6)=0

• The roots (x2+x-6) are 2 and -3 • X=1,2 and -3


38

Examine the nature of roots

• 1.x2-8x+16=0 • Solution • a=1,b=-8 and c=16 • b2-4ac=0 • The roots are real and equal.


39

• 2. 3x2-8x+4=0 • Solution • a=3,b=-8 and c=4 • b2-4ac=16>0 • The roots are real and unequal


40

• 3.5x2-4x+2=0 • Solution • a=5,b=-4 and c=2 • b2-4ac=-24<0 • The roots are imaginary and

unequal


41

Example

• The Value of

• Solution

.........414

14

14

++

++

52±


42

Application of Co-ordinate Geometry

• Co-ordinate Geometry is that branch of mathematics which explains the problems of geometry with the help of algebra.

• 1)The equation of the straight line in simple form

• y= mx +c ,where m is slope and c is a constant


43

• If P(x1,y1) and Q(x2,y2) be the two points on the line then the distance between two points PQ

• PQ= 212

212 )yy()xx( −+−


44

• If P(x1,y1) and Q(x2,y2) be the two points on

the line then the slope • Slope (m) =

12

12

xxyy

−−


45

• A straight line makes that X intercept a’ and Y intercept is b

• Then the equation form is

1=+by

ax


46

• 1.Two lines are having slopes m1 and m2 are parallel to each other if and only if if

m1 =m2

• 2. Two lines are having slopes m1 and m2 are perpendicular to each other if and only if if

m1 .m2=-1


47

4.)The equation ax+by+c=0 be a straight line ,the equation parallel to above line is

ax+ by +k=0 5.The equation ax+by+c=0 be a straight

line ,the equation perpendicular to above line is

b x- by+ k=0


48

• 6.The equation of line passing through the points of intersection of the lines

ax +by +c=0 and a1x +b1y +c=0 can be written as

ax+by+c+K(a1x +b1y +c), where k is constant


49

• 7.The equation of line joining the points (X1,Y1) and (x2,Y2) is given and (x3,y3) on the line then the condition of collinear is

• x1(y2-y3)+x2(y3-y1)+x3(y1-y2)=0


50

Example

• Show that the points A(2,3)B(4,1) and C(-2,7) are collinear

• Solution 2(1-7)+4(7-3)-2(3-1)=-12+16-4=0 Which is true So the given points are collinear


51

Example

• Find the equation of a line passing through the point (5,-4) and parallel to the line 4x+7y+5=0

Solution: The equation parallel to given equation is ax+ by +k=0

• The required equation is 4x+7y+8=0


52

• 1)If ________, the roots are real but unequal

• A) b2 – 4ac = 0 • B) b2 – 4ac >0 • C) b2 – 4ac<0 • D) b2 – 4ac <0


53

• 1)If ________, the roots are real but unequal

• A) b2 – 4ac = 0 • B) b2 – 4ac >0 • C) b2 – 4ac<0 • D) b2 – 4ac <0


54

• 2.The equation of line passing through the points (1, -1) and (3, -2) is given by ________.

• A) 2x + y + 1 = 0 • B) 2x + y + 2 = 0 • C) x + y + 1 = 0 • D) x + 2 y + 1 = 0


55

• 2.The equation of line passing through the points (1, -1) and (3, -2) is given by ________.

• A) 2x + y + 1 = 0 • B) 2x + y + 2 = 0 • C) x + y + 1 = 0 • D) x + 2 y + 1 = 0


56

• 3.The equation –7x + 1 = 5 – 3x will be satisfied for x equal to

• A) 2 • B)-1 • C)1 • D) None of these


57

• 3.The equation –7x + 1 = 5 – 3x will be satisfied for x equal to

• A) 2 • B)-1 • C)1 • D) None of these


58

• 4. The sum of two numbers is 52 and their difference is 2. The numbers are

• A) 17 and 15 • B) 12 and 10 • C) 27 and 25 • D) None of these


59

• 4. The sum of two numbers is 52 and their difference is 2. The numbers are

• A) 17 and 15 • B) 12 and 10 • C) 27 and 25 • D) None of these


60

• 5.Under Algebraic Method we get ______ linear equations

• A) One • B) Two • C) Three • D) Five


61

• 5.Under Algebraic Method we get ______ linear equations

• A) One • B) Two • C) Three • D) Five


62

• 6. The equation of a line passing through (3, 4) and slope 2 is

• A) y – 2x + 2 = 0 • B) y – 3x + 4 = 0 • C) y – 4x + 3 = 0 • D) y – 2x + 4 = 0


63

• 6. The equation of a line passing through (3, 4) and slope 2 is

• A) y – 2x + 2 = 0 • B) y – 3x + 4 = 0 • C) y – 4x + 3 = 0 • D) y – 2x + 4 = 0


64

• 7. The slope of the equation x – y + 5 = 0 is _________.

• A) 1 • B)-1 • C)5 • D)-5


65

• 7. The slope of the equation x – y + 5 = 0 is _________.

• A) 1 • B)-1 • C)5 • D)-5


66

• 8. If one root of the equation x2+ 7x+ p = 0 be reciprocal of the other then the value of p is________.

• A) 1 • B)-1 • C)7 • D)-7


67

• 8. If one root of the equation x2+ 7x+ p = 0 be reciprocal of the other then the value of p is________.

• A) 1 • B)-1 • C)7 • D)-7


68

• 9. Find the distance between the pair of points p (–5, 2) and q (–3, –4)

• A) 2 .Sqrt of10 • B)10. Sqrt of2 • C)2 • D)10


69

• 9. Find the distance between the pair of points p (–5, 2) and q (–3, –4)

• A) 2 .Sqrt of10 • B)10. Sqrt of2 • C)2 • D)10


70

• 10. For what value of 'K' the equation 9x2 – 24x + K = 0 has equal roots

• A) –16 • B)-15 • C)0 • D)16


71

• 10. For what value of 'K' the equation 9x2 – 24x + K = 0 has equal roots

• A) –16 • B)-15 • C)0 • D)16

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