Answer:
The only options which are having same solution as the given equation is A, B and D.
Explanation:
Given the equation: \(\frac{3}{5}x+ \frac{2}{3} +x =\frac{1}{2}- \frac{1}{5}x\) ......[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
\(\frac{3x+5x}{5}+ \frac{2}{3} =\frac{1}{2}- \frac{1}{5}x\)
Simplify:
\(\frac{8x}{5}+\frac{2}{3} =\frac{1}{2}- \frac{1}{5}x\)
or we can write this as;
\(\frac{8}{5}x +\frac{2}{3} =\frac{1}{2}- \frac{1}{5}x\) .
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;
\(\frac{9x+10+15x}{15} = \frac{5-2x}{10}\)
By cross multiplication we get;
\(10 \cdot (9x+10+15x) = 15 \cdot (5-2x)\)
Divide both side by 5 we get;
\(2 \cdot (9x+10+15x) = 3 \cdot (5-2x)\)
Using distributive property on both sides of an equation (i.e, \(a\cdot (b+c) =a\cdot b+a\cdot c\) )
we have;
\(18x+20+30x = 15-6x\) ......[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];
\(18x+20+30x+6x = 15-6x+6x\)
Simplify;
\(18x+20+30x+6x = 15\)
Combine 18x and 6x we get;
\(24x+20+30x= 15\) ......[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;
\(24x+20+30x-20= 15-20\)
Simplify:
\(24x+30x= -5\)
Therefore, from the given options only A, B and D have the same solution as \(\frac{3}{5}x+ \frac{2}{3} +x =\frac{1}{2}- \frac{1}{5}x\)
