You can put this solution on YOUR website!
Let quantity of 5% acid required be x
Let quantity of 6.5% acid required be y
Mixture = 6%
Equate percentages
5%x+6.5%y=6%*200
multiply by 100
5x+6.5y=1200................(1)
Quantity
x+y=200.....................(2)
multiply equation (2) by -5
we get -5x-5y=-1000
Add this equation to (1)
we get
1.5y=200
y= 200/1.5
y=133.3 ml
x= balance amount= 66.67ml

You can put this solution on YOUR website!
let rate of wind be x m/h
speed against wind 550-x m/h----------- 2000 miles
speed with wind = 550+x m/h------------2400 miles
time = d/r
time against = time with
2000/(550-x)= 2400/(550+x)
2000(550+x)=2400(550-x)
1100000+2000x=1320000-2400x
2400x+2000x=1320000-1100000
4400x=220000
x= 50 m/h

You can put this solution on YOUR website!
Let the car and motorcycle meet after distance x
car speed = 80 km/h
motorcycle speed = 90 km/h
They both travel the same distance
The car takes 4 minutes more .= 4/60 = 1/15 hours
time = distance /speed
car time - motorcycle time = 1/15 hours
(x/80)-(x/90) = 1/15
LCD = 720
multiply equation by 720
9x-8x= 720/15
x= 48
They will meet after 48 km

You can put this solution on YOUR website!


/6

Find the last term of perfect square
Last term = 1/2 the coefficient of 2nd. term squared
(1/2*1/6)^2
(1/12)^2= 1/144
add (1/44) to both sides

( taking the LCD

Take the square root of both sides
(z+(1/12)) = +/- (7/12)
z= -(1/12) +(7/12) --->1/2
OR
z= -(1/12) -(7/12) ---> -2/3

You can put this solution on YOUR website!
Y=-1/2ax-5,m=5/2
What is the question?
Anyway I think we have to find a
y= 1/2 ax
compare with y= mx
so m= 1/2a
5/2 = 1/2 a
a= 5

You can put this solution on YOUR website!
(3,7) slope 2
equation of line =
y-y1=m(x-x1) where (x1,y1) is the point on the line
y-7=2(x-3)
y-7=2x-6
y-2x=1
y-2x=1 is the required equation

You can put this solution on YOUR website!
plane speed x mph
wind speed 25 mph

Distance with wind 700 miles
Distance against wind 600 miles

speed with wind x + 25 mph
speed against x -25 mph
Time with wind= 700 /( x + 25 )
time against wind 600 / ( x -25 )

Time with wind = time against

700/(x+25)=600/(x-25)
divide by 100
7/(x+25)=6/(x-25)
7(x-25)=6(x+25)
7x-175=6x+150
7x-6x=175+150
x=325
speed of plane = 325 mph

You can put this solution on YOUR website!
Let x be the number of miles
Ford 26 + 0.16 x
chevrolet 23 + 0.18 x
Ford > chevrolet
26 + 0.16 x > 23 + 0.18 x
add -26 to both sides
26+0.16n-26>23 +0.18x-26
0.16 n- -23 > > 0.18 x -3
add -0.18 x
-0.02 x > -3
/ -0.02
x < 150
x should be less than 150 for Ford to be a better deal
OR Chevrolet will be expensive after 150 miles

You can put this solution on YOUR website!
y=-2x,2y=x,4y=2x+4
Line l, y=-2x,slope (m) = -2
line p, 2y=x, slope (m) = 1/2
line n, 4y=2x+4 slope (m) = 1/2
line l is perpendicular line p & n
line p & line n are parallel
If the slopes are equal with different y intercepts then the lines are parallel.
if the product of the slopes is = -1 then the lines are peripendicular.

You can put this solution on YOUR website!
let the total investment be x
25x% @ 8%
40% @ 10.5% bonds
(x-(0.25x+0.4x) @ 13%
0.25x**8%+0.4x*10.5%+ (1-0.25x+0.4x)*0.13% = 4300
multiply by 100
8*0.25x+10.5*0.4x+0.35x*13=4300*100
2x+4.2x+4.55x=430000
10.75x=430000
x=430000/10.75
= 40,000
Total invested = $40,000
25% of 40,000@ 8% --> $10,000
40% of 40,000 @ 10.5% bonds ----->$ $16000
35% of 40,000 @ 13%---->$14000

You can put this solution on YOUR website!

You can put this solution on YOUR website!
using the formula (y-y1)=m(x-x1) where (x1,y1) are the co ordinates of the point on the line
The point is (3,0)--- (x1,y1)
slope = 4/5
y-0= 4/5(x-3)
multiply by 5
5y=4(x-3)
5y=4x-12
/5
y=(4/5)x-(12/5)

You can put this solution on YOUR website!
Part I 7.00% per annum ------------- Amount invested =x
Part II 2.00% per annum ------------ Amount invested = y =
1400
Interest----- 58

Part I 7.00% per annum ---x
Part II 2.00% per annum ---y
Total investment
x + y= 1400 -------------1
Interest on both investments
7.00% x + 2.00% y= 58
Multiply by 100
7 x + 2 y= 5800.00 --------2
Multiply (1) by -7
we get
-7 x -7 y= -9800.00
Add this to (2)
0 x -5 y= -4000
divide by -5
y = 800
Part I 7.00% $ 600
Part II 2.00% $ 800

CHECK
600 --------- 7.00% ------- 42.00
800 ------------- 2.00% ------- 16.00
Total -------------------- 58.00

m.ananth@hotmail.ca

You can put this solution on YOUR website!
Part I 9.00% per annum ------------- Amount invested =x
Part II 2.00% per annum ------------ Amount invested = y =
2300
Interest----- 95

Part I 9.00% per annum ---x
Part II 2.00% per annum ---y
Total investment
x + y= 2300 -------------1
Interest on both investments
9.00% x + 2.00% y= 95
Multiply by 100
9 x + 2 y= 9500.00 --------2
Multiply (1) by -9
we get
-9 x -9 y= -20700.00
Add this to (2)
0 x -7 y= -11200
divide by -7
y = 1600
Part I 9.00% $ 700
Part II 2.00% $ 1600

CHECK
700 --------- 9.00% ------- 63.00
1600 ------------- 2.00% ------- 32.00
Total -------------------- 95.00

m.ananth@hotmail.ca

You can put this solution on YOUR website!
a + 9 b = 1 .............1

a + 3 b = -5 .............2
Eliminate y
multiply (1)by 1
Multiply (2) by -3
1 a 9 b = 1
-3 a -9 b = 15
Add the two equations
-2 a = 16
/ -2
a = -8
plug value of a in (1)
1 a + 9 b = 1
-8 + 9 b = 1
9 b = 1 + 8
9 b = 9
b = 1
a -8
b 1
m.ananth@hotmail.ca

You can put this solution on YOUR website!
4^2X=64
2x log 4 = log 64
2x= log 64/log4
2x=3
x= 3/2

You can put this solution on YOUR website!
x = log72/log6 = 2.3869.
x= 2.39

You can put this solution on YOUR website!
let distance be x
passenger train speed = 60 mi/h
time pasanger train = x/60
Similarly time freight train = x/40
Difference between the two timings is 1.5
x/40 -x/60 = 1.5
LCD = 120
3x-2x=120*1.5
x= 180 miles
Passenger train takes 180/60 = 3 hours
Freight tain takes 180/40 = 4.5 hours

You can put this solution on YOUR website!
y= 3x -8

Assume values of x and plug in the equation to get values of y.
Plot the points
x 0 5 2
y= -8 7 -2
(x,y)(0,-8),(5, 7),(2-2)
6 x + -2 y = 8
y= 3 x -4
Assume values of x and plug in the equation to get values of y.
Plot the points
x 0 5 2
y= -4 11 2
(x,y) (0,-4),(5,11),( 2,2)


The slopes are same. The lines are parallel. Hence no solution to the system of equations
m.ananth@hotmail.ca

You can put this solution on YOUR website!
x1 y1 x2 y2
-8 8 -3 -6

slope m = (y2-y1)/(x2-x1)
( -6 - 8 )/( -3 - (-8) )
( -14 / 5 )
m= -14/5

You can put this solution on YOUR website!
x1 y1 x2 y2
-8 8 -3 -6

slope m = (y2-y1)/(x2-x1)
( -6 - 8 )/( -3 - (-8) )
( -14 / 5 )
m= -14/5

You can put this solution on YOUR website!



Take the square root
(a+1)=+/-5
a=-1+5=4
OR
a=-1-5=-6
a=4OR-6

You can put this solution on YOUR website!
A(-3,1)B(1,4)C(4,0) D(0,-3)
Find the slopes of AB,BC,CD,DA
Slope of AB
x1 y1 x2 y2
-3 1 1 4

slope m = (y2-y1)/(x2-x1)
( 4 - 1 )/( 1 - -3 )
( 3 / 4 )
m(AB)= 3/4
M(BC)
x1 y1 x2 y2
4 0 1 4

slope m = (y2-y1)/(x2-x1)
( 4 - 0 )/( 1 - 4 )
( 4 / -3 )
m= -4/3
m(CA)
x1 y1 x2 y2
4 0 0 -3

slope m = (y2-y1)/(x2-x1)
( -3 - 0 )/( 0 - 4 )
( -3 / -4 )
m= 3/4
m(AD)
x1 y1 x2 y2
-3 1 0 -3

slope m = (y2-y1)/(x2-x1)
( -3 - 1 )/( 0 - -3 )
( -4 / 3 )
m= -4/3

Two lines are having same slopes and two are having slopes negative reciprocal
So the angles formed are 90 degrees.
If the distances aer equal then the co ordinates form a square.
l(AB)
x1 y1 x2 y2

-3 1 1 4
d=
d=
d=
d=
d= 5
l(BC)
x1 y1 x2 y2

4 0 1 4
d=
d=
d=
d=
d= 5

l(CD)
x1 y1 x2 y2

4 0 0 -3
d=
d=
d=
d=
d= 5
l(DA)
x1 y1 x2 y2

-3 1 0 -3
d=
d=
d=
d=
d= 5
The lengths are all 5 units. and they are at right angles. so it is a square.

You can put this solution on YOUR website!
percent ---------------- quantity
Acid solution I 12 ---------------- x liters
Acid solution II 20 ------ 4 - x liters
Mixture 15 ---------------- 4

12x+20(4-x)=15*4
1200x+2000(4-x) =6000
1200x+8000-2000 x=6000
1200x- 2000x=6000 -8000
-800 x = -2000
/ -800
x=2.5 liters 12.00% Acid solution I
1.5 liters 20.00% Acid solution II

m.ananth@hotmail.ca

You can put this solution on YOUR website!
$4500--------------9.25%
x............13.5%
Total yield = 12%
4500*9.5%+13.5%(x)=12%*(4500+x)
multiply by 100
4500*9.5+13.5(x)=12(4500+x)
42750+13.5x=154000+12x
13.5x-12x=54000-42750
1.5x=11250
x= 7500

$ 7500 has to be invested to get a total return of 12%

You can put this solution on YOUR website!
Part I 11.20% per annum ------------- Amount invested =x
Part II -4.70% per annum ------------ Amount invested = y
18000
Interest----- 1062

Part I 11.20% per annum ---x
Part II -4.70% per annum ---y
Total investment
x + 1 y= 18000 -------------1
Interest on both investments
11.20% x + -4.70% y= 1062
Multiply by 100
11.2 x + -4.7 y= 106200.00 --------2
Multiply (1) by -11.2
we get
-11.2 x -11.2 y= -201600.00
Add this to (2)
0 x -15.9 y= -95400
divide by -15.9
y = 6000
Part I 11.20% $ 12000
Part II -4.70% $ 6000

CHECK
12000 --------- 11.20% ------- 1344.00
6000 ------------- -4.70% ------- -282.00
Total -------------------- 1062.00

m.ananth@hotmail.ca

You can put this solution on YOUR website!
we solve by the squaring method
When we add 64 then we get a perfect square on the left side of the equation



Take the square root on both sides
(x+8) = +/-
x= -8+ OR x=-8-

You can put this solution on YOUR website!
Part I 11.20% per annum ------------- Amount invested =x
Part II -4.70% per annum ------------ Amount invested = y
18000
Interest----- 1062

Part I 11.20% per annum ---x
Part II -4.70% per annum ---y
Total investment
x + 1 y= 18000 -------------1
Interest on both investments
11.20% x + -4.70% y= 1062
Multiply by 100
11.2 x + -4.7 y= 106200.00 --------2
Multiply (1) by -11.2
we get
-11.2 x -11.2 y= -201600.00
Add this to (2)
0 x -15.9 y= -95400
divide by -15.9
y = 6000
Part I 11.20% $ 12000
Part II -4.70% $ 6000

CHECK
12000 --------- 11.20% ------- 1344.00
6000 ------------- -4.70% ------- -282.00
Total -------------------- 1062.00

m.ananth@hotmail.ca

You can put this solution on YOUR website!
8%-------------------x
9%-------------------2x
Balance @
5%-------------------(5500-3x)
Interest = 449
0.08x+0.09(2x)+0.05(5500-3x)=449
Multiply by 100
8x+18x+27500-15x=44900
11x=44900-27500
11x=17400
x= 1581.82................8%
2x=3163.64.................9%
754.55.....................5%

You can put this solution on YOUR website!
Boat speed =x mph
current speed =y mph
againstcurrent x-y 4 hours
with current x+y 2 hours

Distance = 60 miles distance= 60
t=d/r against current
60.00 / ( x - y )= 4.00

4.00 x -4.00 y = 60.00 ....................1
downstream
60.00 / ( x + y )= 2.00
2.00 ( x + y ) = 60.00
2.00 x + 2.00 y = 60.00 ...............2
Multiply (1) by 1.00
Multiply (2) by 2.00
we get y
4.00 x -4.00 y = 60.00
4.00 x + 4.00 = 120.00
8.00 x = 180.00
/ 8.00
x = 22.5 mph

plug value of x in (1) y
4 x -4 y = 60
90 -4 -90 = 60
-4 y = 60
-4 y = -30 mph
y = 7.5
Boat 22.5 mph
current 7.5 mph

You can put this solution on YOUR website!
2y<4
First the equality
2y=4
y=2
It is a line parallel to the x axis
The equality

Since y is < than something the line will be dotted one.the shaded portion will be below the line
The inequality