Write an equation: three sisters went shopping for mother’s day. each sister bought a gift for their mom. maggie spent 3 times as much as karen. karen spent half as much as jasmine. altogether, they spent $60. then, solve your equation to determine how much each sister spent on their gift. describe what happens when an equation has no solution. provide an example and explain how it represents your description. describe what happens when an equation has infinite solutions. provide an example and explain how it represents your description. i'll mark the brainliest : )

Karen spent $ 10 , Jasmine spent $20 , Maggie spent $30

When x disappears from the equation and the two sides of the equation not equal each other , then the equation has no solution

When x disappears from the equation and the two sides of the equation equal each other , then the equation has infinite solutions

Step-by-step explanation:

* Lets read the problem and solve it

- There are three sisters ⇒ Maggie , Karen , Jasmine

- Each sister bought a gift for their mom

- Maggie spent 3 times as much as Karen

- Karen spent half as much as Jasmine

- Altogether, they spent $60

- Let Jasmine spent x

∵ Karen spent half as much as Jasmine

∴ Karen spent 1/2 x

∵ Maggie spent 3 times as much as Karen

∴ Maggie spent 3 × 1/2 x = 3/2 x

∵ Altogether, they spent $60

∴ x + 1/2 x + 3/2 x = 60

∴ 3x = 60 ⇒ divide both sides by 3

∴ x = 20

* Lets find the share of each one

∵ Jasmine spent x

∴ Jasmine spent $20

∵ Karen spent 1/2 x

∴ Karen spent 1/2 × 20 = $10

∵ Maggie spent 3/2 x

∴ Maggie spent 3/2 x 20 = $30

* Karen spent $ 10 , Jasmine spent $20 , Maggie spent $30

* Lets describe what happens when an equation has no solution

- "No solution" means that there are no values of x satisfy the equation

- When x disappears from the equation and the two sides of the

 equation not equal each other , then the equation has no solution

- Ex: solve the equation 2 + 2x – 8 = 7x + 5 – 5x

∵ 12 + 2x – 8 = 7x + 5 – 5x ⇒ add the like terms

∴ 4 + 2x = 2x + 5 ⇒ subtract 2x from both sides

∴ 4 = 5 ⇒ the two sides not equal

∴ There is no value of x satisfy this equation

∴ There is no solution

* Lets describe what happens when an equation has infinite solutions

- "Infinite solutions" means all numbers are solutions.

- When x disappears from the equation and the two sides of the

 equation equal each other , then the equation has infinite solutions

- Ex: solve the equation 2x + 3 = x + x + 3

∵ 2x + 3 = x + x + 3 ⇒ add the like terms

∴ 2x + 3 = 2x + 3 ⇒ subtract 2x from both sides

∴ 3 = 3 ⇒ the two sides are equal

∴ All values of x satisfy the equation

∴ There are infinite solutions

the value of v.w= -12.

step-by-step explanation:

we are given three vector r,v and w as:

r = < 5, -5, -2> ; v = < 2, -8, -8> ; w = < -2, 6, -5>

we know that r,v and w could also be represented as:

r=5i-5j-2k

v=2i-8j-8k

and w= -2i+6j-5k

where i,j and k are the unit vectors of x,y and z-axis respectively.

also we know that it has the following property:

i.i=1 ; j.j=1; k.k=1

and i.=j.i=i.k=k.i=j.k=k.j=0

i.e. when we multiply two vectors or we find it's dot product we need to only multiply the components of x coordinates with each other; y with each other and z with each other.

hence, v.w= (2i-8j- -2i+6j-5k)

                  = 2×(-2)+(-8)×(6)+(-8)×(-5)

                  = -4-48+40

                  = -12

hence, v.w= -12