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

Option (a) , (b) and ( d) are equivalent to given expression
`(3)/(5)x+(2)/(3)+x= (1)/(2)- (1)/(5)x`

Explanation:

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

We have to use properties to rewrite the given equation and check which are correct from thee given options,

Consider the given equation,

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

Applying commutative property of addition ,
`a+b=b+ a`

Equation becomes,

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

Now adding x terms on right side , we get,

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

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

Thus, obtained option (a).

Again consider given equation ,

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

Taking LCM both sides, we get,

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

Solving , we get,

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

Cross multiply, we get,

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

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

Thus, obtained option (b).

Taking like terms together,

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

`\Rightarrow 24x+30x=-5`

Thus, obtained option (d).

Thus, Option (a) , (b) and ( d) are equivalent to given expression
`(3)/(5)x+(2)/(3)+x= (1)/(2)- (1)/(5)x`

Answer 2

we have

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

Combine like terms in both sides

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

we know that

`((3)/(5)x+ x)=((3)/(5)x+ (5)/(5)x)=(8)/(5)x`

substitute in the expression above

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

Multiply equation A by
`5*3*2=30` both sides

`30*((8)/(5)x+(2)/(3))=30*((1)/(2)-(1)/(5)x)`

`48x+20=15-6x` ---------> equation B

Group terms that contain the same variable, and move the constant to the opposite side of the equation

`48x+6x=15-20`

`54x=-5` ---------> equation C

Solve for x

`x=-(5)/(54) =-0.09`

We are going to proceed to verify each case to determine the solution.

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

the case a) is equal to the equation A

so

the case a) have the same solution that the given equation

Case b)
`18x+20+30x=15-6x`

Combine like terms in left side

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

`(48x)+20=15-6x`

the case b) is equal to the equation B

so

the case b) have the same solution that the given equation

Case c)
`18x+20+x=15-6x`

Combine like terms in left side

`(18x+x)+20=15-6x`

`(19x)+20=15-6x`

`19x+6x=15-20\\25x=-5\\x=-0.20`

`-0.20\\eq -0.09`

therefore

the case c) not have the same solution that the given equation

Case d)
`24x+30x=-5`

Combine like terms in left side

`54x=-5`

the case d) is equal to the equation C

so

the case d) have the same solution that the given equation

Case e)
`12x+30x=-5`

Combine like terms in left side

`42x=-5`

`x=-5/42=-0.12`

`-0.12\\eq -0.09`

therefore

the case e) not have the same solution that the given equation

therefore

the answer is

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

case b)
`18x+20+30x=15-6x`

case d)
`24x+30x=-5`