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

we have

$\frac{3}{5}x+ \frac{2}{3}+x=\frac{1}{2}-\frac{1}{5}x$

Combine like terms in both sides

$(\frac{3}{5}x+ x)+\frac{2}{3}=\frac{1}{2}-\frac{1}{5}x$

we know that

$(\frac{3}{5}x+ x)=(\frac{3}{5}x+ \frac{5}{5}x)=\frac{8}{5}x$

substitute in the expression above

$\frac{8}{5}x+\frac{2}{3}=\frac{1}{2}-\frac{1}{5}x$-----> equation A        

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

$30*(\frac{8}{5}x+\frac{2}{3})=30*(\frac{1}{2}-\frac{1}{5}x)$

$48x+20=15-6x$ ---------> 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

$x=-\frac{5}{54} =-0.09$

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

Case a) $\frac{8}{5}x+\frac{2}{3}=\frac{1}{2}-\frac{1}{5}x$

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

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

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

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

$-0.20\neq -0.09$

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

$42x=-5$

$x=-5/42=-0.12$

$-0.12\neq -0.09$

therefore

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

therefore

the answer is

case a) $\frac{8}{5}x+\frac{2}{3}=\frac{1}{2}-\frac{1}{5}x$

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

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

Answer 2

Answer:

Option (a) , (b) and ( d) are equivalent to given expression $\frac{3}{5}x+\frac{2}{3}+x= \frac{1}{2}- \frac{1}{5}x$

Step-by-step explanation:

 Given equation :  $\frac{3}{5}x+\frac{2}{3}+x= \frac{1}{2}- \frac{1}{5}x$

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

Consider the given equation,

$\frac{3}{5}x+\frac{2}{3}+x= \frac{1}{2}- \frac{1}{5}x$

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

Equation becomes,

$\frac{3}{5}x+x+\frac{2}{3}= \frac{1}{2}- \frac{1}{5}x$

Now adding x terms on right side , we get,

$\frac{3+5}{5}x+\frac{2}{3}= \frac{1}{2}- \frac{1}{5}x$

$\Rightarrow \frac{8}{5}x+\frac{2}{3}= \frac{1}{2}- \frac{1}{5}x$

Thus, obtained option (a).

Again consider given equation ,

$\frac{3}{5}x+\frac{2}{3}+x= \frac{1}{2}- \frac{1}{5}x$

Taking LCM both sides, we get,

$\frac{9x+10+15x}{15}= \frac{5-2x}{10}$

Solving , we get,

$\frac{9x+10+15x}{3}= \frac{5-2x}{2}$

Cross multiply, we get,

$2\times (9x+10+15x)=3\times(5-2x)$

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

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 $\frac{3}{5}x+\frac{2}{3}+x= \frac{1}{2}- \frac{1}{5}x$