Question

Use properties to rewrite the given equation. Which equations have the same solution as 3/5x +2/3 + x = 1/2– 1/5x? Check all that apply.

a. 8/5x+2/3=1/2-1/5x
b. 18x + 20 + 30x = 15 – 6x
c. 18x + 20 + x = 15 – 6x
d. 24x + 30x = –5
e. 12x + 30x = –5

Answer 1

Only options a ,b,d have the same solution as
`(3)/(5)x +(2)/(3)+x=(1)/(2)-(1)/(5)x`

Explanation:

Given:

An equation:
`(3)/(5)x +(2)/(3)+x=(1)/(2)-(1)/(5)x` ....[1]

Like terms states that contain the same variables raised to the same power.

Combine like terms on left hand side in equation [1] we get,

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

`(8)/(5)x +(2)/(3)=(1)/(2)-(1)/(5)x`

we get option (a) as the same solution.

Using LCM(Least Common ) to change each fraction to make their denominators the same as the least common denominator

Taking LCM both sides in Equation (1) ,

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

As LCM(3,5)=15 and LCM(2,5)=10 we have,

`(9x+10+15x)/(15)= (5-2x)/(10)`

On simplifying we get,

`10(9x+10+15x)= 15(5-2x)`=

Dividing both sides by 5 we get,

`(10(9x+10+15x))/(5) =(15(5-2x))/(5)`

On simplify:

`2(9x+10+15x)=3(5-2x)`

∴
`18x+20+30x=15-6x`

which is same solution as given in option (b).

Now,

Use Additive Property of Equality states that allows one to add the same quantity to both sides of an equation.

Further, using the above property add 6x both sides in
`18x+20+30x=15-6x` we get,

`18x+20+30x+6x=15-6x+6x`

On simplifying we get,

`18x+20+30x+6x=15`

Subtract 20 from both the sides we get,

`18x+20+30x+6x-20=15-20`

Simplify:

`18x+30x+6x=-5`

or,
`24x+30x=-5` which is same solution as given in option(d)

But, options c and e equation doesn't have same solution as
`(3)/(5)x +(2)/(3)+x=(1)/(2)-(1)/(5)x`