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50 paise coins = x
25 paise coins = 2x
She has Rs34 = 3400 paise
50*x+25(2x)=3400
50x+50x=3400
100x=3400
x=3400/100
x= 34 ---- 50 paise coins
25 paise coins ---- 34*2= 68
CHECK
68*25 + 34*50 = 3400
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Karen speed = x
time she traveled = 6 hours
so she traveled 6x miles
Sister speed = x+5 mph
time she traveled = 4 hours
so she traveled 4(x+5) miles
4(x+5)+6x=600
4x+20+6x=600
10x=580
x=580/10
x= 58 mph
58*6=348
63*4=252
Total = 600
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Boat speed x mph
current speed 5 mph

Distance with current 36 miles
Distance against current 12 miles

spee with current x + 5 mph
spee against current= x -5 mph
Total time = 9 hours
Time with current= 36 /( x + 5 )
time against current 12 / ( x -5 )

Time with current + time against = 9 hours

36/(x + 5 ) + 12 /(x -5 ) = 9
LCD = ( x + 5 )* (x -5 )
multiply the equation by the LCD
we get
3*(x -5)+ 12 (x +5)= 9 (x^2 -25 )
36)x -180 + 12 x + 60 ) = 9 X^2 -225
48 x -120 = 9 X^2 -225
9 X^2 -48 x 120 + -225 = 0
9 X^2+ -48 x+ -105 = 0
/ 9
1 X^2 -5.33 x -11.67 = 0

Find the roots of the equation by quadratic formula

a= 1 ,b= -5.33 ,c= -11.67

b^2-4ac= 28.41 + 46.68
b^2-4ac= 75.09
49.98


x1=( 5.33 + 8.67 )/ 2
x1= 7 7
x2=( 5.33 -8.67 ) / 2
x2= -1.67
Ignore negative value
Boat speed = 7 mph

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Let x be the number of shirts
Francisco R, C(f)= 50 + 4.25 x
Gary Ray, C(g)= 20 + 6.70 x
Francisco R < Gary Ray C(r) 20 + 6.70 x < 50 + 4.25 x
20 + 6.70 x < 50 + 4.25 x
20 - 50 < 4.25 x -6.7 x
-30 < -2.45 x
12.24 < x
The quantity should be greater than 13 shirts
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Cross multiply
(2x+1)(2x+3)=(4x+6)(x-3)
2x(2x+3)+1(2x+3)=4x(x-3)+6(x-3)
4x^2+6x+2x+3=4x^2-12x+6x-18
14x=-21
x=-21/14
x=-3/2
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length = x
width = (x-12)
Area = L*W
Area = x(x-12)=540
x^2-12x=540
x^2-12x-540=0
x^2-30x+18x-540=0
x(x-30)+18(x-30)=0
(x-30)(x+18)=0
x=30 which is positive
length = 30 ft
width = 18 feet
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Thin candle burning rate is 30/3 = 10 cm/h
Thick candle burning rate = 30/5 = 6 cm/h
Thin candle burns 10-6 = 4 cm/h faster than thick candle
Thicker candle is lit at 7
thinner candle is lit at 7.30
so difference = 1/2 hour.
in 1/2 hour the thin candle would have burnt 10*1/2 = 5 cm
time both will have same height = difference in length when second candle is lighted / difference between burning rates
5/4 = 1.25 hours
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Consider the equation y= mx+b
Since the line bisects two Quadrants It passes through the origin.
the y intercept is =0
so the equation will be y=x
but the line will be from left to right sloping downwards passing the IV quadrant.
so it will have negative slope
The equation will be y=-x

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m= 8

Plug value of the slope and point ( 6 , 7 ) in
Y = m x + b
7.00 = 48 + b
b= 7 - 48
b= -41
So the equation will be
Y = 8 x -41
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5a-9a= -4a

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Dave x mph
Kent 5 faster = x+ 5

Dave 135 miles
Kent 150 miles
Dave time = Kent time
t=d/r
135 / x = 150 /( x + 5 )
cross multiply
135 ( x + 5 ) = 150 x
135 x + 675 = 150 x
150 x - 135 x = 675
15 x = 675
/ 15
x = 45 mph Dave

50 mph Kent
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x1 y1 x2 y2
-4 -8 8 4

slope m = (y2-y1)/(x2-x1)
( 4 - -8 )/( 8 - -4 )
( 12 / 12 )
m= 1

Plug value of the slope and point ( -4 , -8 ) in
Y = m x + b
-8.00 = -4 + b
b= -8 - -4
b= -4
So the equation will be
Y = x -4

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the LCD of the equation is 12
multiply each term by 12
12y+2=4(x-4)
12y+2=4x-16
-4x+12y=-18
4x-12y=18
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--------price ---------------- quantity
Coffee I 90 ---------------- 50 lbs
coffee II 60 ---------------- x lbs
Mixture 70 ---------------- 50 + x lbs

90*50+60 x=70(50 +x)
4500+60 x=3500 + 70 x
60 x + -70 x = 3500 - -4500
-10 x = -1000
/ -10
x = 100
100 lbs of 60 % coffee II

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--------percent ---------------- quantity
Tin I 70 ---------------- 70 Kgs
Tin II 30 ---------------- x Kgs
Mixture 50 ---------------- 70 + x Kgs

70*70+30 x= 50 ( 70 + x )
4900 + 30x = 3500 + 50 x
30 x + -50 x = 3500 - -4900
-20 x = -1400
/ -20
x = 70
70 Kgs of 30 % Tin II

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14 x + 12 y = 562 .............1
1 x + 1 y = 42 .............2
Eliminate y
multiply (1)by 1
Multiply (2) by -12
14 x 12 y = 562
-12 x -12 y = -504
Add the two equations
2 x = 58
/ 2
x = 29
plug value of x in (1)
14 x + 12 y = 562
406 + 12 y = 562
12 y = 562 -406
12 y = 156
y = 13

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0.5x + 2.2y = 9
6x + 0.4y = -22
multiply both equations by 10
5x+22y=90
60x+4y=-220
5 x + 22 y = 90 .............1
60 x + 4 y = -220 .............2
Eliminate y
multiply (1)by -4
Multiply (2) by 22
-20 x -88 y = -360
1320 x 88 y = -4840
Add the two equations
1300 x = -5200
/ 1300
x = -4
plug value of x in (1)
5 x + 22 y = 90
-20 + 22 y = 90
22 y = 90 + 20
22 y = 110
y = 5

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let the number be x
twice the number = 2x
2x- 1/3 x =30
multiply by 3
6x-x=90
5x=90
/5
x=18

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x(x-2)^3-35(x-2)^2=0
x(x-2)^3=35(x-2)^2
divide by (x-2)^2
x(x-2)=35
x^2-2x=35
x^2-2x-35=0
x^2-7x+5x-35=0
x(x-7)+5(x-7)=0
(x-7)(x+5)=0
x=7 OR -5

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(2,5) slope =4
y=mx+b
plug slope and point
5=8+b
b=-3
so the equation will be
y=4x-3

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x1 y1 x2 y2
-1 7 -2 -15

slope m = (y2-y1)/(x2-x1)
( -15 - 7 )/( -2 - -1 )
( -22 / -1 )
m= 22

Plug value of the slope and point ( -1 , 7 ) in
Y = m x + b
7.00 = -22 + b
b= 7 - -22
b= 29
So the equation will be
Y = 22 x + 29
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p^2-25/p^2-10p+25



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--------percent ---------------- quantity
glycerin 50 ---------------- 60 gallons
water 0 ---------------- x gallons
Mixture 10 ---------------- 60 + x gallons

50 * 60 + 0 x = 10 ( 60 + x )
3000 + 0 x = 600 + 10 x
0 x + -10 x = 600 - -3000
-10 x = -2400
/ -10
x = 240
240 gallons of water

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y>6x+2
First determine the boundary of the inequality
The boundary
y=6x+2
when x=0 y=2 (0,2)
when y=0, x=-2/3 =-1/3 (-1/3,0)

Since y > than something the upper side of the line will be shaded. The line will be dotted


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square I Area = 150

=
perimeter =
Perimeter =
square II
area =54
side =
side =
perimeter =
Add them up
Total perimeter =
=cm
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let the foot of the ladder be located at M
let the distance on ground from ladder foot to base of building A be MA
let the distance on ground from ladder foot to base of building B be MB
MA^2= 13^2-12^2
MA^2= 169-144=25
MA=5 m
...
MB^2= 13^2-8^2
MB^2=169-64
MB^2= 105
MB= 10.25 m
MA + MB = alley width
5+10.25= 15.25 m

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diameter of well = 14 m, radius = 7
depth = 20 m
Volume of mud dug out = pi*r^2*h
V= 3.14*7^2*20
V= 3077.2 m^3
Area of Land = 240* 180 = 43200 sq. m
Volume of the land after filling = L*W*H
dug out mud is the volume increase
3077.2 = 43200*H
H=3077.2/43200
H= 0.0712 m
0.0712 *100 = 7.12 cm
The height will increase by 7.12 cm
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daughter = x years
mom=9x
after 5 years
daughter = x+5 years
Mom = 9x+5 years
1/5(9x+5)=x+5
9x+5= 5(x+5)
9x+5=4x+25
9x-4x=25-5
5x=20
/4
x=4 daughter's age
mom's age = 9*4 = 36 years
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x+2y=2...........1
2x+4y=36...........2

equation1
when x= 0 y= 1 ( 0 , 1 )
when y = 0 x= 2 ( 2 , 0 )
Equation 2
when x= 0 y= 9 ( 0 , 9 )
when y = 0 x= 18 ( 18 , 0 )

You will observe that the graphs have same slope and different intercepts.
So the lines are parallel to each other.
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let me try
3(5+1)
This means 3 x (5+1)
3 x 6
=18
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one number be x
Other number will be x+4
The product = x(x+4)
x(x+4)=4+(x+x+4)
x^2+4x=2x+8
x^2-2x+4x-8=0
x^2+2x-8=0
x^2+4x-2x-8=0
x(x+4)-2(x+4)=0
(x+4)(x-2)=0
x=-4 OR 2
The numbers are (-4,0) , (2,6)
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