16801 equations

EQUATIONS

Quantitative Aptitude & Business Statistics

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.
Quantitative Aptitude & Business
Statistics: Equations
2

Simple Equations
• A simple equation is one unknown X
is in the form aX+b=0
• Where and b are known constants .

Quantitative Aptitude & Business 3

Example
• Solve for X
4x 14 19
−1 = x+
3 15 5

• By transposing the variables in one side the
constants in other side ,we have
4 x 14 x 19
− = +1
3 15 5

Solution

( 20 − 14) x 19 + 5
=
15 5
6x 24
=
15 5
24 × 15
X = = 12
5× 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


Solution
• Let X be the numerator and the
Fraction be
• X/X+5
• By the x+3 3
=
x+5+3 4
Question or 4 x + 12 = 3 x + 24
X = 12


• The required Fraction
12
17


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 .


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.

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


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

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


Simultaneous Linear Equations in
two unknowns
• Methods of Solution
• 1.Method of elimination

• 2Cross Multiplication.


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.


Example
• Solve 2x+5y=9 and 3x-y=5

• x=2
• y=1


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


X Y 1
= =
(b1c2 − b2 c1 ) (c1 a2 − c2 a1 ) (a1b2 − a2 b1 )
b1c2 − b2 c1 c1a 2 − c2 a1
X= ;Y =
a1b2 − a 2 b1 a1b2 −a 2 b

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


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 ;
− b ± b − 4ac
2
X=
2a


Sum and product of roots
• Let the roots of equation be α and β

b
• Sum of roots ∞+β =−
a

• Products of roots ∞β = c
a


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


Nature of the Roots

− b ± b − 4ac 2
X=
2a
• 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

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


Example
• Solve
• X2-5x+6=0 a=1,b=-5 and c= 6

− b ± b − 4ac
2
X=
2a

• X=3 and 2

Example
• If α and β be the roots of
X2+7x+12=0
• Find the equation whose roots are
•(α + β ) 2 and (α − β ) 2


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

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

α2 β2
+
β α


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

α β α + β (α + β ) − 3αβ
2 2 3 3 3
+ = = = −22
α β αβ αβ


Example
• Solve
x+2
4 − 3.2
x
+2 =0 5


Solution
x+2
4 − 3.2
x
+2 =0 5

( 2 ) − 3.2 2 + 32 = 0
2 x x 2

( 2 ) − 3.2 2 + 32 = 0
x 2 x 2

y − 12 y + 32 = 0
2


Solution
• (y-8)(y-4)=0
• y=8 or y=4

• 2x=8 or 2x=4

• X=3 or 2

Example
• If one root of the equation is 2 − 3
form the equation
• Solution
• If one root is 2 − 3 then the other
roots 2 + 3
• Sum of roots =4
• Product of roots =1

Solution
• Required equation is
• x2-x(Sum of roots ) +Product of
roots=0
• x2-4x+1=0


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

1 1 1
+ =
x 25 − x 6
x − 25 x + 150 = 0
2

( x − 15)( x − 10) = 0

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


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 -3Quantitative Aptitude & Business 37

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.


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


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

Example
• The Value of 1
4+
1
4+
1
4+
4 + .........

• Solution
2± 5

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 Quantitative Aptitude & Business
42

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

• PQ= ( x 2 − x 1 )2 + ( y 2 − y 1 )2


• If P(x1,y1) and Q(x2,y2) be the two points on
the line then the slope
y 2 − y1
• Slope (m) =
x 2 − x1


• A straight line makes that X
intercept a’ and Y intercept is b
• Then the equation form is
x y
+ =1
a b


• 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


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


• 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

• 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


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


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

• 1)If ________, the roots are real but
unequal
• A) b2 – 4ac = 0
• B) b2 – 4ac >0
• C) b2 – 4ac<0
• D) b2 – 4ac <0


• 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

• 3.The equation –7x + 1 = 5 – 3x will be
satisfied for x equal to
• A) 2
• B)-1
• C)1
• D) None of these


• 3.The equation –7x + 1 = 5 – 3x will be
satisfied for x equal to
• A) 2
• B)-1
• C)1


• 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


• 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


• 5.Under Algebraic Method we get ______
linear equations
• A) One
• B) Two
• C) Three
• D) Five


• 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


• 7. The slope of the equation x – y + 5 = 0
is _________.
• A) 1
• B)-1
• C)5
• D)-5


• 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

• 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


• 10. For what value of 'K' the equation 9x2
– 24x + K = 0 has equal roots
• A) –16
• B)-15
• C)0
• D)16


16801 equations

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