Answer:
The only options which are having same solution as the given equation is A, B and D.
Explanation:
Given the equation: [tex]\frac{3}{5}x+ \frac{2}{3} +x =\frac{1}{2}- \frac{1}{5}x[/tex] ......[1]
Like terms are terms whose variables are the same.
In the given equation [1],
Combine like terms of variable x on LHS; we have
[tex]\frac{3x+5x}{5}+ \frac{2}{3} =\frac{1}{2}- \frac{1}{5}x[/tex]
Simplify:
[tex]\frac{8x}{5}+\frac{2}{3} =\frac{1}{2}- \frac{1}{5}x[/tex]
or we can write this as;
[tex]\frac{8}{5}x +\frac{2}{3} =\frac{1}{2}- \frac{1}{5}x[/tex] .
LCM(Least Common Multiple) to change each fraction to make their denominators the same as the least common denominator.
Taking LCM to both sides of an equation [1] as;
LCM of 3, 5 on LHS is 15 and LCM of 2 , 5 on RHS is 10;
[tex]\frac{9x+10+15x}{15} = \frac{5-2x}{10}[/tex]
By cross multiplication we get;
[tex]10 \cdot (9x+10+15x) = 15 \cdot (5-2x)[/tex]
Divide both side by 5 we get;
[tex]2 \cdot (9x+10+15x) = 3 \cdot (5-2x)[/tex]
Using distributive property on both sides of an equation (i.e, [tex]a\cdot (b+c) =a\cdot b+a\cdot c[/tex] )
we have;
[tex]18x+20+30x = 15-6x[/tex] ......[2]
Additive Property of Equality states that allows one to add the same quantity to both sides of an equation.
Using additive property of equality:
Add 6x to both sides of an equation in [2];
[tex]18x+20+30x+6x = 15-6x+6x[/tex]
Simplify;
[tex]18x+20+30x+6x = 15[/tex]
Combine 18x and 6x we get;
[tex]24x+20+30x= 15[/tex] ......[3]
Subtraction Property of Equality states that allows one to subtract the same quantity to both sides of an equation.
Subtract 20 from both sides of an equation in [3] we get;
[tex]24x+20+30x-20= 15-20[/tex]
Simplify:
[tex]24x+30x= -5[/tex]
Therefore, from the given options only A, B and D have the same solution as [tex]\frac{3}{5}x+ \frac{2}{3} +x =\frac{1}{2}- \frac{1}{5}x[/tex]
