LurpDaDerp

x + y = 10

x^3 + y^3 = 162 + x^2 + y^2

x^3 + y^3 factors into (x+y)(x^2+xy+y^2)

we know x+y is 10 from the first equation so we plug that in

so now we have: 

10(x^2+xy+y^2) = 162+x^2+y^2

expanding gives us

10x^2+10xy+10y^2 = 162+x^2+y^2

combine like terms

9x^2 + 9y^2 + 10xy - 162 = 0

divdie both sides by 9

x^2 + y^2 + 9/10xy - 18 = 0

Since x+y = 10, (x+y)^2 = 100 and x^2+y^2 = 100 - 2xy

plugging that in gives

100-2xy+9/10xy-18=0

combine like terms

82-11/10xy=0

11/10xy = 82

xy = 820/11

now knowing the sum and product of x and y, we can create quadratic satisfying these properties using vieta's and solve 

x+y = -b/a

xy = c/a

let a = 1 for convenience

we get 

n^2 -10n + 820/11 = 0

solving gives

x and y = 5 + i(sqrt(545/11)) and 5 - i(sqrt(545/11))

first find midpoint of line

(((x₁+x₂)/2),((y₁+y₂)/2))

= (((3-1)/2),((2+7)/2))

= (1,9/2)

equation of original line

1. find slope (7-2)/(-1-3) = -5/4

2. write in point slope form: y - 2= -5/4(x - 3)

3. expand into slope intercept form: y = -5/4x + 23/4

find slope of perpendicular line, which is -1/m, where m is original slope

-1/(-5/4) = 4/5

equation of perpendicular line passing through (1,9/2): y = 4/5x + 23/4

Combine x terms and isolate x

2x+7x+x <= 12+18+5

10x <= 35

x<=35/10

x<=7/2

expand and combine like terms first

-4x-16 > 9x+28

Isolate the x

-44 > 13x

-44/13 > x

x < -44/13 or (-∞,-44/14)

Make an equation system

Let Grogg's current age = x

Let Grogg's dad's current age = y

8(x-7) = y-7

5(x-3) = y-3

subtract equations from each other

8(x-7) - 5(x-3) = -4

solve for x

8x-56 - 5x+15 = -4

3x = 37

x = 37/3

plug in for y

8(37/3-7) = y-7

y = 149/3

Make an equation

x + (x+1) + (x+2) + (x+3) + (x+4) + (x+5) + (x+6) = 21*4(x+6)

(i'm assuming that when you say 21*4 you mean multiplication, but I'm a bit confused because no one writes problems like that and this does not give interger solutions) 

Solving gives: 

7x + 21 = 84x + 504

77x = -483

x = -483/77 ???

There must be a mistype in the problem or there is something wrong in my work, but there are no interger solutions for x. 

if 21*4 were to refer to 4 then the solution would be x = 1, solving the same way. 

Make an equation system

Let Grogg's current age = x

Let Grogg's dad's current age = y

8(x-7) = y-7

5(x-3) = y-3

subtract equations from each other

8(x-7) - 5(x-3) = -4

solve for x

8x-56 - 5x+15 = -4

3x = 37

x = 37/3

plug in for y

8(37/3-7) = y-7

y = 149/3

a + ab = 250

a - ab = -240

Method 1: 

Add the two equations so that the ab's cancel out to get: 

2a = 10

so a = 5

Plug into equations to check: 

5 + 5b = 250

1 + b = 50

b = 49 

5 - 5b = -240

1 - b = -48

b = 49

In both cases, b remains the same, therefore 5 works as an answer

Method 2: 

Graphing (💀💀💀)

(i was too lazy to graph by myself)

but yes, a = 5

Firstly add 111 to every number on the list so that the list becomes 544, 540, 536, ... , 8, 4. Then divide everything by 4 to get 136, 135, 134, ... , 1. 

This list becomes 1, 2, 3, 4, ... , 135, 136. So, there are 136 numbers in this list.