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For each function, find f(-2), f(-1/2), f(3), and f(7).

f(x)= -1/3x + 1
f(-2)=-1/3(-2)+1
f(-2)=1/6+1
f(-2)=(1+6)/6
f(-2)=7/6


f(x)= -1/3x + 1
f(-1/2)=-1/(3(1/-2))+1
f(-1/2)=-1/(3/-2)+1
f(-1/2)=2/3+1
f(-1/2)=(2+3)/3
f(-1/2)=5/3


f(x)= -1/3x + 1
f(3)=-1/3(3)+1
f(3)=-1/9+1
f(3)=(-1+9)/9
f(3)=8/9



f(x)= -1/3x + 1
f(7)=-1/3(7)+1
f(7)=-1/21+1
f(7)=(-1+21)/21
f(7)=20/21

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What number is 62% of 25
=62% * 25
=62/100 * 25
=62/4
=15.5
15.5 is 62% of 25

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Ally A
Amount = x
Concentration = 25%=0.25
================================================================================
Alloy B
Amount = 45-x pounds
Concentration = 62%=0.62
================================================================================
Resultant Alloy
Amount=45
Concentration=34%=0.34
================================================================================
(Amount of A * Concentration of A)+(Amount of B * Concentration of B) = (Amount of Resultant * Concentration of Resultant)
(x*0.25)+(45-x)(0.62)=(45)*0.34
0.25x+27.9-0.62x=15.3
-0.37x=15.3-27.9
-0.37x=-12.6
-0.37x/-0.37=-12.6/-0.37
x=34.05 pounds
============================
Ally A
Amount = 34.04 pounds
Concentration = 25%=0.25
================================================================================
Alloy B
Amount = 45-x=45-34.04=10.95 pounds
Concentration = 62%=0.62

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find the total cost of an engagement ring at 3240 including sale tax of 6.5%?
6.5% of 3240 = 3240*6.5%=3240*(6.5/100)=3240*0.065=210.6
Total cost = 3240+210.6=3450.60


or
find 106.5% of 3240 = 3240*(106.5/100)=3240*1.065=3450.6

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Soluton A
Amount = 10 ml
Concentration = 30%=0.3
================================================================================
Solution B
Amount = x ml
Concentration = 100%=1 (Pure acid)
================================================================================
Resultant Solution
Amount=10+x ml
Concentration=50%=0.5
================================================================================
(Amount of A * Concentration of A)+(Amount of B * Concentration of B) = (Amount of Resultant * Concentration of Resultant)
(10*0.3)+(x*1)=(10+x)*0.50
3+x=5+0.5x
x-0.5x=5-3
0.5x=2
0.5x/0.5=2/0.5
x=4 ml
================================================================================
10 ml of a 30% acid solution and 4 ml of pure acid must be mixed to obtain 14 ml of 50% solution

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(3, 12) and (15, -2)
Slope of line = Change in y / change in x
m=y2-y1/x2-x1
m=-2-12/15-3
m=-14/12
m=-7/6
Slope = -7/6

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Standard Form of Equation of the line:
y=mx+b
Given line
y=5x
Compare above equation with the standard form equation
Slope = m=5 and y-intercept = b=0
Since parallel lines have same slope so slope of the required line will be:-
m=5
Now we have a point(2,-1) and slope (5)of the line we can easily find required lines by putting these values in the equation of the straight line poin-slope form.
m=(y2-y1)/(x2-x1)
5=(y-(-1))/(x-2)
5=(y+1)/(x-2)
5(x-2)=y+1
5x-10=y+1
y+1=5x-10
y=5x-10-1
y=5x-11
Above equation is the required equation of the line that is parallel to y=5x and contains (2,-1)

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Soluton A
Amount = x
Concentration = 40%=0.4
================================================================================
Solution B
Amount = 96-x liters
Concentration = 25%=0.25
================================================================================
Resultant Solution
Amount=96
Concentration=27%=0.27
================================================================================
(Amount of A * Concentration of A)+(Amount of B * Concentration of B) = (Amount of Resultant * Concentration of Resultant)
(x*0.4)+(96-x)(0.25)=(96)*0.27
0.4x+24-0.25x=25.92
0.15x=25.92-24
0.15x=1.92
0.15x/0.65=1.92/0.15
x=12.8 liters

================================================================================
Soluton A
Amount = x = 12.8 liters
Concentration = 40%=0.4
================================================================================
Solution B
Amount = 96-x=96-12.8=83.2 liters
Concentration = 25%=0.25
===============================================================================
12.8 liters of 40% concentration and 83.2 liters of 25% concentration solutoins must be mixed to obtain 96 liters of a 27% solution

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Bus A Data
==========
Speed = 58mph
Time=t hour
Distance Covered by Bus A=58*t
=============
Bus B Data
============
Speed = 52mph
Time=t hour
Distance covered by Bus B=52*t
Distance Covered by Bus A + Distance Covered by Bus B = Total Distnace apart
52t+58t=363
110t=363
110t/110=363/110
t=3.3 hour

They will be 363 miles apart after 3.3 hours or 3 hours 18 minutes.


Check
=============
Bus A Data
==========
Speed = 58mph
Time=3.3 hours
Distance Covered by Bus A=58*3.3=191.4 miles
Bus B Data
============
Speed = 52mph
Time=3.3 hours
Distance covered by Bus B=52*3.3 = 171.6 miles
Total Distance between two Buses = 191.4+171.6 = 363 miles

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A car left seattle traveling 45mph. A second car left from the same place one hour later on the same road traveling 60mph. How long will it take the second car to catch up with the first car. Which of these is the correct second eqaution for the problem?

Car A Data
==========
Speed = 45mph
Time = t hour
Distance covered = x miles
Distance = speed*time
x=45*t
x=45t.............(1)

Car B Data
==========
Speed = 60mph
Time =t-1(hour)
Distance covered = x
Distance = Speed * time
x=60*(t-1)

Both cars will cover same distance
45t=60*(t-1)
45t=60t-60
45t-60t=60
-15t=-60
-15t/-15=-60/-15
t=4 hours
Car A travel time 4 hours and Car B travel time 3 hours.
After 3 hours of leaving same place Car B will catch Car A
Check
========
Car A Data
==========
Speed = 45mph
Time = 4 hour
Distance covered = x miles
Distance = speed*time
x=45*4
x=180 miles
Car B Data
==========
Speed = 60mph
Time =t-1=4-1=3(hour)
Distance covered = x
Distance = Speed * time
x=60*3
x=180 miles

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Let x and y are two numbers

x:y=7:11
x/y=7/11
11x=7y
x=7y/11............(1)

(x+7):(y+7)=2:3
(x+7)/(y+7)=2/3
3(x+7)=2(y+7)
3x+21=2y+14
3x=2y+14-21
3x=2y-7............(2)
Put the value of x in (2) from (1)
3(7y/11)=2y-7
21y/11=2y-7
21y=11*(2y-7)
21y=22y-77
21y-22y=-77
-y=-77
y=77
Put the value of y in (1)
x=7y/11............(1)
x=7(77)/11
x=7*7
x=49


Numbers are 49 and 77


49/77=7/11=7:11
(49+7)/(77+7)=56/84=28/42=14/21=2/3=2:3

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0.3x-0.2y=4...............(1)
0.2x + 0.3y=29/19.........(2)
Make above equations more simple
3x/10+2y/10=4
Multiply by 10 both sides of above equation
3x+2y=40.................(3)
2x/10+3y/10=29/19
Multiply by 10 both sides of above equation
2x+3y=290/19.............(4)
Multiply (3) by -2 and (4) by 3 and add the resultant equations
-6x-4y=-80
6x+9y=870/19
------------------
5y=870/19-80
5y=(870-1520)/19
5y=-650/19
5y/5=-650/19/5
y=-130/19
Put the value of y in (3)
3x+2y=40.................(3)
3x+2(-130/19)=40
3x-260/19=40
Multiply by 19 both sides
57x-260=760
57x=760+260
57x=1020
57x/57=1020/57
x=1020/57
x, y =1020/57,-130/19
Check
Put the values of x and y in (4)
2x+3y=290/19.............(4)
2(1020/57)+3(-130/19)=290/19
2040/57-390/19=290/19
(2040-1170)/57=290/19
870/57=290/19
290/19=290/19

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Length = l=w+5
Width = w=?
Perimeter = p = 9w
==============================
Perimeter of a Rectangle = length + length + width + width
Perimeter of a Rectangle =2 length +2 width
P = 2l + 2w
9w=2(w+5)+ 2w
9w=2w+10+2w
9w=4w+10
9w-4w=10
5w=10
5w/5=10/5
w=2
============================
Dimensions of the Rectangle
Length = l=w+5 = 2+5 =7
Width = w=2
Perimeter = p = 9w = 9*2=18

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4(2x-3)=-2(3x-8)
8x-12=-6x+16
8x+6x=16+12
14x=28
14x/14=28/14
x=2
Check
4(2x-3)=-2(3x-8)
4(2(2)-3)=-2(3(2)x-8)
4(4-3)=-2(6-8)
4(1)=-2(-2)
4=4

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1.)
x + 2y = -1............(1)
4x - 5y = 22...........(2)
Multiply (1) by -4 and add to (2)
-4x-8y=4
4x-5y=22
-13y=26
-13y/-13=26/-13
y=-2
Put the value of y in (1)
x+2y=-1
x+2(-2)=-1
x-4=-1
x=-1+4
x=3
x,y = 3, -2
Check
Put the values of x and y in (1)
x + 2y = -1............(1)
3+2(-2)=-1
3-4=-1
-1=-1

2.)
-3x + 7y = 14 ..........(1)
2x - y = -13...........(2)
Multiply (2) by 7 and add to (2)
-3x + 7y = 14
14x - 7y = -91
--------------
11x=-77
11x/11=-77/11
x=-7
Put the value of x in (2)
2x - y = -13...........(2)
2(-7)-y=-13
-14-y=-13
-y=-13+14
-y=1
y=-1
x,y=-7,-1
Check
Put the values of x and y in (2)
2(-7) - (-1) = -13
-14 + 1 = -13
-13=-13

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Perimeter = P = 40 ft
Width = w
Length = w+2
Perimeter = 2l +2w
40=2(w+2)+2w
40=2w+4+2w
40=4w+4
40-4=4w
36=4w
36/4=4w/4
9=w
w=9
Perimeter = P = 40 ft
Width = w = 9
Length = w+2 = 9+2 = 11

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x + y = 6 ...........(1)
x - y = -2 ..........(2)
Add (1) and (2)
x + y = 6
x - y = -2
------------
2x=4
2x/2=4/2
x=2
Put the value of x in (1)
x + y = 6 ...........(1)
2+y=6
y=6-2
y=4
x,y=2,4

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A man is 4 times as old as his son. Five years ago, the sum of their ages was 30.Form an equation in a variable of your choice and hence calculate their present ages.
Let
Present age of Man = m
Present age of Son = s


Man is 4 times as old as his son.
m=4s..........(1)
Five years ago,
m-5=s-5
the sum of their ages was 30
(m-5)+(s-5)=30
m-5+s-5=30
m+s-10=30
m+s=30+10
m+s=40.............(2)
Put the value of m from (1) to (2)
4s+s=40
5s=40
5s/5=40/5
s=8
Put the value of s in (1)
m=4s
m=4(8)
m=32
Present age of Man = m = 32 years old
Present age of Son = s = 8 years old

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6y-2+2y=14
8y-2=14
8y=14+2
8y=16
8y/8=16/8
y=2

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Let
Number = x
(4+x)/3=5
(4+x)=5*3
4+x=15
x=15-4
x=11
Number is 11

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Saidhari bought a toaster oven for 90 dollars. this price was 1/4 off the list price. what was the list price.
List price = x
1/4 of list price = x(1/4)=x/4
List pirce - discount = Discounted price
x-x/4=90
4x-x/4=90
3x/4=90
x=90*4/3
x=30*4
x=120
List price = $120

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solve for r, 36r^2-1=0
36r^2-1=0
(6r)^2-(1)^2=0
(6r-1)(6r+1)=0
(6r-1)=0 or(6r+1)=0
6r=1 or6r=-1
r=1/6 or r=-1/6

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You are right well done!
Keep it up!!
solve for x
ax-x=c
x(a-1)=c
x(a-1)/(a-1)=c/(a-1)
cancel out (a-1) from left hand side of the equation
x=c/(a-1)

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Let it takes x weeks to equalize their savings.
15*x=4100+10*x
15x=4100+10x
15x-10x=4100
5x=4100
5x/5=4100/5
x=820
After 820 weeks or 205 months or 17 years, one month and one week !!!!!

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6x=7-4y
4y=-6x+7
4y/4=(-6x+7)/4
y=(-6x+7)/4
=====================
x=-2
y=(-6(-2)+7)/4
y=(12+7)/4
y=19/4
===================
x=-1
y=(-6(-1)+7)/4
y=(6+7)/4
y=13/4
=================
x=0
y=(-6(0)+7)/4
y=7/4

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4x=3y-7; x= a, 5, 6
4x=3y-7
rearrange above equation
3y-7=4x
3y=4x+7
3y/3=(4x+7)/3
y=(4x+7)/3
===========================
Now
x=a
y=(4a+7)/3
==========================
x=5
y=(4(5)+7)/3
y=(20+7)/3
y=27/3
y=9
==========================
x=6
y=(4(6)+7)/3
y=(24+7)/3
y=24/3+(7/3)
y=8+(7/3)

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Let
Present age of Father = f
Present age of Son = s

Two years ago
f-2=s-2
Father was 3 times as old as his son
f-2=3(s-2)
f-2=3s-6
f=3s-6+2
f=3s-4............(1)
and 2 year hence,
f+2=s+2
then twice his age will be equal to 5 times that of his son
2(f+2)=5(s+2)
2f+4=5s+10
2f=5s+10-4
2f=5s+6.............(2)
Put the value of f in (2) from (1)
2(3s-4)=5s+6
6s-8=5s+6
6s-5s=6+8
s=14
Put the value of s in (1)
f=3s-4............(1)
f=3(14)-4
f=42-4
f=38
Present age of Father = f = 38 years old
Present age of Son = s = 14 years old

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Let
Present age of Rico = r
Present age of Ryan = y


Rico is twice old as Ryan
r=2y........(1)
In four years,Rico's age will be three times as Ryan's age is six years ago
Rico age in four years = r+4
Ryan age six years ago = y-6
r+4=3(y-6)
r+4=3y-18
r=3y-18-4
r=3y-22.............(2)
Put the value of r in (2) from (1)
2y=3y-22
2y-3y=-22
-y=-22
Multiply by -1 both sides
y=22
Put the value of y in (1)
r=2y
r=2(22)
r=44
Present age of Rico = r = 44 years old
Present age of Ryan = y = 22 years old
Six years ago Ryan was = 22-6 = 16
In four years Rico will be = 44+4 = 48 (which is 3 times of 16)

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Let
Present age of Anna = a
In 36 years
a+36=5a
a+36=5a
a-5a=-36
-4a=-36
-4a/-4=-36/-4
a=9
Anna is 9 years old now.
In 36 years she will be =(36+9)= 45 (which is 5 times of 9)

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Let
Present age of Selma = s
Present age of Joyce = j

selma is now 3 times as old as joyce
s=3j........(1)
Four years ago,
s-4=j-4
then
Selma was 4 times as old as joyce
s-4=4(j-4)
s-4=4j-16
s=4j-16+4
s=4j-12.............(2)
Put the value of s in (2) from (1)
3j=4j-12
3j-4j=-12
-j=-12
Multiply by -1 both sides
j=12
Put the value of j in (1)
s=3j
s=3(12)
s=36
Present age of Selma = s = 36 years old
Present age of Joyce = j = 12 years old

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In 1985
========
Maria was = m years old
Sue was = s years old
Maria was three times as old as Sue
m=3s........(1)
1980
====
Five years earlier
(s-5)+(m-5)=18
s-5+m-5=18
s+m-10=18
s+m=18+10
s+m=28.......(2)
=============================================================================
Put the value of m from (1) to (2)
s+(3s)=28
s+3s=28
4s=28
4s/4=28/4
s=7
Put the value of s in (1)
m=3s
m=3(7)
m=21
In 1985
Maria was = m = 21 years old
Sue was = s = 7 = years old
Maira was born in = 1985 - 21 = 1964