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Density = mass/ volume
density = mass/ (area * tk.)
2.6989= 1.47/(305*tk)
(305*tk)= 1.47/2.6989
tk = 1.47/(2.6989*305)
thickness = 0.001785 cm

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The equation of line with in intercept form is




multiply equation by LCD -12
so 4x-3y=-12
3y=4x-12
y = 4/3 x -4
slope = m= 4/3
Perpendicular line will have slope which is negative reciprocal of 4/3
= -3/4
The line is passing through (-3,0)
equation of line will be
(y-y1) = m(x-x1)
y-(-3)=-3/4(x-0)
y+3= -3/4 x
y= -3/4 x -3

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Plane speed =x mph
wind speed =y mph
against wind 5 hours
with wind 4 hours

Distance with wind 4800 miles distance against wind 4800
t=d/r against wind (x-y)
4800.00 / ( x - y )= 5.00

5.00 x - -5.00 y = 4800.00 ....................1
with wind (x+y)
4800.00 / ( x + y )= 4.00
4.00 ( x + y ) = 4800.00
4.00 x + 4.00 y = 4800.00 ...............2
Multiply (1) by 4.00
Multiply (2) by 5.00
we get 2.00
20.00 x + -20.00 y = 19200.00
20.00 x + 20.00 y = 24000.00
40.00 x = 43200.00
/ 40.00
x = 1080 mph

plug value of x in (1) y
5 x -5 y = 4800
5400 -5 -5400 = 4800
-5 y = 4800
-5 y = -600 mph
y = 120
Plane 1080 mph
wind 120 mph

CHECK
x+y= 1200
X-y= 960 +
4800.00 / ( 1200.00 )= 4
4800 / ( 960 ) = 5

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We solve by substitution method
-4 x + 1 y = -11 .............1
1 y = -11 + 4 x
/ 1 =
y = -11 + 4 x
-5 x + -2 y = 10 .....................2
Substitute y in (2)
-5 x + -2 ( -11 + 4 x ) = 10
-5 x + 22 + -8 x = 10
-5 x -8 x = -12
-13 x = -12
/ -13
x= 12/13
Plug the value of x in (1)
-4 x + 1 y = -11
-4 * 12/13 + 1 y = -11
-48/13 + 1 y = -11
1 y = -95/13
y= -95/13
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The equation of a line passing through the points (x1,y1) & (x2,y2) is



(x1,y1) -(5/12,-1), (x2,y2)- (-3/4,1/6)
plug the values in the equation







7/ 6 y-7/ 6 = -7/ 6 x - 35/72

7/ 6 y= -7/ 6 x+49/72
/ 7/ 6
y=-1x+ 7/12
m=-1

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6*12=72
split 72 such that their sum is -14 and product is -72





(r-3)(6r+4)

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Volume of brick = 0.25m * 0.12m * 0.55m = 0.0165 cu.m
volume of 1 brick = 0.0165 cu.m
density = 1.8*10^3 kg/m^3
weight of 1 brick = 0.0165* 1.8*10^3
29.7 kg
15000 bricks will weigh 15000*29.7
=445500 kg
= 445.5 tons
number of trips = 445.5/3 = 1485 trips

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The kite is vertically above the ground. height ?
The length of sting = hypotenuse= 80m
The angle of elevation = 24 deg.
Sin theta = opposite side/ hypotenuse
sin 24 = opposite side/80
height of kite = 80 * sin 24
use calc to find the height
.....
Point A elevation is 67 deg
point B elevation = 48 deg.
height of mountain = 780 m
A to mountain foot distance = 780/Tan 67 .............. (1)
B to mountain foot distance = 780/tan 48 ..................(2)
distance AB = (2)-(1)
780/1.11 -780/2.35
about 370.8 m is the length of the tunnel.

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-3 x + 4 y = -8
Find the slope of this line

4 y = 3 x -8
Divide by 4
y = 3/ 4 x + -2
Compare this equation with y=mx+b, m= slope & b= y intercept
slope m = 3/4

The slope of a line perpendicular to the above line will be the negative reciprocal -4/3
Because m1*m2 =-1
The slope of the required line will be -4/3

m= -4/3 ,point ( -12 , 7 )
Find b by plugging the values of m & the point in
y=mx+b
7 = 16 + b
b= -9
m= -4/3
The required equation is y = -4/ 3 x -9
m.ananth@hotmail.ca

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PARALLEL


4 x -2 y = 3
Find the slope of this line
make y the subject
-2 y = -4 x + 3
Divide by -2
y = 2 x -1 1/2
Compare this equation with y=mx+b
slope m = 2
The slope of a line parallel to the above line will be the same
The slope of the required line will be 2
m= 2 ,point ( 2 , 1 )
Find b by plugging the values of m & the point in
y=mx+b
1 = 4 + b
b= -3
m= 2
Plug value of the slope and b in y = mx +b
The required equation is y = 2 x -3
Perpendicular
4 x + -2 y = 3
Find the slope of this line

-2 y = -4 x + 3
Divide by -2
y = 2 x + -1 1/2
Compare this equation with y=mx+b, m= slope & b= y intercept
slope m = 2

The slope of a line perpendicular to the above line will be the negative reciprocal - 1/2
Because m1*m2 =-1
The slope of the required line will be - 1/2

m= - 1/2 ,point ( 2 , 1 )
Find b by plugging the values of m & the point in
y=mx+b
1 = -1 + b
b= 2
m= - 1/2
The required equation is y = - 1/ 2 x 2
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y= -2 x -1

Assume values of x and plug in the equation to get values of y.
Plot the points
x 0 1 2 -2
y= -1 -3 -5 3
(x,y) (0,-1),(1,-3),( 2,3)

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allison is saving money to buy a video game system. in the first week, her savings were $8 less then 2/5 of the price of the system. in the second week, she saved 50 cents more than 1/2 the price of the system. she was still $37 short. whats the price of the system?
Let system cost $x
First week -- (2/5)x -8
II week-------(1/2)x +0.5
She was$37 short
Add up
savings + short = price of system
(2/5)x-8 + (1/2)x+0.5 +37 =x
(2/5)x +(1/2)x -x= -37-0.5+8
(4x+5x-10x)/10 = -29.5
-x = 10*-29.5
-x=-295
x= $ 295 the price of the system

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Part I 5.00% per annum ------------- Amount invested =x
Part II 9.00% per annum ------------ Amount invested = y
6000
Interest----- 380

Part I 5.00% per annum ---x
Part II 9.00% per annum ---y
Total investment
x + 1 y= 6000 -------------1
Interest on both investments
5.00% x + 9.00% y= 380
Multiply by 100
5 x + 9 y= 38000.00 --------2
Multiply (1) by -5
we get
-5 x -5 y= -30000.00
Add this to (2)
0 x 4 y= 8000
divide by 4
y = 2000
Part I 5.00% $ 4000
Part II 9.00% $ 2000

CHECK
4000 --------- 5.00% ------- 200.00
2000 ------------- 9.00% ------- 180.00
Total -------------------- 380.00

m.ananth@hotmail.ca

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-------------- percent ---------------- Amount
Acid I 3 ---------------- 2 quarts
Acid II 10 ---------------- x quarts
Mixture 4 ---------------- 2 + x quarts

3 * 2 + 10 x = 4 ( 2 + x )
6 + 10 x = 8 + 4 x
10 x -4 x = -6 + 8
6 x = 2
/ 6
x = 0.33 quarts Acid II

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Find the roots of the equation by quadratic formula

a= 2 b= -3 c= -11

b^2-4ac= 9 - -88
b^2-4ac= 97
= 9.85


x1=( 3 + 9.85 )/ 4
x1= 3.21
x2=( 3 -9.85 ) / 4
x2= -1.71

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a(b + c) = ab + ac
you multiply each term in the parenthisis with the term outside it.
example
2h+2(2h+4)=26
2h+(2*2h)+(2*4)=26
2h+4h+8=26
6h+8 =26
-8
6h=18
/6
h=3

4 x 53
(4 x 50) + (4 x 3)
200 + 12
212

4 x 27
=4(20 +7)
=4(20) + (7)
=80 + 28
=108
The Distributive Property
a(b + c) = ab + ac
3x(x+6)
= 3x(x) + 3x(6)
=3x2+18x
(x + y)(x + 2y)
x(x+2y) + y(x+2y)

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-------------- percent ---------------- Amount
solution I 73 ---------------- 143 liters
solution II 37 ---------------- x liters
Mixture 51 ---------------- 143 + x liters

73 * 143 + 37 x = 51 ( 143 + x )
10439 + 37 x = 7293 + 51 x
37 x -51 x = -10439 + 7293
-14 x = -3146
/ -14
x = 224.71 liters solution II

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Part I 4.00% per annum ------------- Amount invested =x
Part II 6.00% per annum ------------ Amount invested = y
8000
Interest----- 380

Part I 4.00% per annum ---x
Part II 6.00% per annum ---y
Total investment
x + 1 y= 8000 -------------1
Interest on both investments
4.00% x + 6.00% y= 380
Multiply by 100
4 x + 6 y= 38000.00 --------2
Multiply (1) by -4
we get
-4 x -4 y= -32000.00
Add this to (2)
0 x 2 y= 6000
divide by 2
y = 3000
Part I 4.00% $ 5000
Part II 6.00% $ 3000

CHECK
5000 --------- 4.00% ------- 200.00
3000 ------------- 6.00% ------- 180.00
Total -------------------- 380.00

m.ananth@hotmail.ca

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x= number of child tickets
y= number of adult tickets
Total sold
1.00 x + 1.00 y = 150.00 .............1
Total value
4.75 x + 7.50 y = 891.25 .............2
Eliminate y
multiply (1)by -7.50
Multiply (2) by 1.00
-7.50 x -7.50 y = -1125.00
4.75 x + 7.50 y = 891.25
Add the two equations
-2.75 x = -233.75
/ -2.75
x = 85.00
plug value of x in (1)
1.00 x + 1.00 y = 150.00
85.00 + y = 150.00
y = 150.00 -85.00
y = 65.00
y = 65.00
x= 85 number of child tickets
y= 65 number of adult tickets
m.ananth@hotmail.ca

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20x+3y=20 and -20x+5y=60 using elimination
20x+3y=20 -20x+5(10)=60
-20x+5y=60 -20x+50= 60
----------- -20x= -50 +60
0+8y=80 --- ---
-- -- -20x= 10
8 8 ---- ---
y=10 x=-0.5

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Find Amount
Principal P = 3700
Amount= A
years=n 5.00
compounded 1 times a year t
Rate = 6.00 0.06
Amount = P*((n+r)/n)^n*t

Amount = = 3700 *( 1 + 0.06 /t)^ 5 * 1
Amount = 3700 *( 1 + 0.06 )^ 5
3700 *( 1.06 )^ 5
Amount = 4951.43

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prime factors of 81 are 3*3*3*3

3*3*3*3=81
so 3^4= 81
3^*=4=3^4
*=4

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take the equation y=mx+b
if m is positive the line will rise from the third quadrant to the first quadrant.
if m is negative the line will rise from the fourth quadrant to the second quadrant.
When the value of | b | the y intercept increases the angle made by the line with X axis increase.

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x^4=1
x^4-1=0
(x^2+1)(x^2-1)=0
(x^2+1)(x+1)(x-1)=0
If (x^2+1)=0
x^2=-1
x= sqrt(-1)=i
OR x= -1 OR +1

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sinx = 1+cos^2x
sinx = 1+1-sin^2x
sin x= 2-sin^2x
Rearrage
sin^2x+sinx-2 =0
factor the quadratic equation
sin^2x+2sinx-sinx-2=0
sinx(sinx+1)-1(sinx+2)=0
(sinx+1)(sinx-1)=0
sin x= -1 OR xinx =1

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=(3x+2)

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I = P * r * t
P= 8000
r=2% = 0.02
t=?
I = 400
400=8000*0.02*t
400/(8000*0.02) =t
400/160=t
2.5 = t
2 1/2 years

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10 x + 2 y = 46 .............1

3 x -3 y = 21 .............2
Eliminate y
multiply (1)by 3
Multiply (2) by 2
30 x 6 y = 138
6 x -6 y = 42
Add the two equations
36 x = 180
/ 36
x = 5
plug value of x in (1)
10 x + 2 y = 46
50 + 2 y = 46
2 y = 46 -50
2 y = -4
y = -2
x= 5, y= -2
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1 y = -3 x -7
Divide by 1
y = -3 x + -7
Compare this equation with y=mx+b, m= slope & b= y intercept
slope m = -3

The slope of a line perpendicular to the above line will be the negative reciprocal 1/3
Because m1*m2 =-1
The slope of the required line will be 1/3

m= 1/3 ,point ( 3 , -2 )
Find b by plugging the values of m & the point in
y=mx+b
-2 = 1 + b
b= -3
m= 1/3
The required equation is y = 1/ 3 x -3
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4 x + 1 y = 10 .............1
1 y = 10 + -4 x
/ 1 =
y = 10 + -4 x
14 x + -1 y = -1 .....................2
Substitute y in (2)
14 x + -1 ( 10 + -4 x ) =
14 x + -10 + 4 x = -1
14 x 4 x = 9
18 x = 9
/ 18
x= 1/ 2
Plug the value of x in (1)
4 x + 1 y = 10
4 * 1/ 2 + 1 y = 10
2 + 1 y = 10
1 y = 8
/ 1
y= 8
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let number be x
reciprocal = 1/x
reciprocal of 3
=1/3
(1/3)-x
(1/x) + (1/(x-3) = (8/x)
multiply equation by x(x-3)
(x-3) +x=8(x-3)
x-3+x=8x-24
6x=21
x=7/2