Question

How to solve and respond.

Answer 1

Step-by-step explanation:

the error:

it is stated that

$\frac{5x}{x+7} +\frac{7}{x} = \frac{5x}{x+7} +\frac{7(7)}{x+7}$

subtract (5x)/(x+7) from both sides

$\frac{7}{x} = \frac{7(7)}{x+7}$

multiply both sides by x + 7

7(x+7) = 49x

7x + 49 = 49x

subtract 7x from both sides to isolate x and its coefficient

49 = 42x

thus, this is only true when 49 = 42x. in order for these two equations to be equal, they must always be true, so this is wrong

the solution:

we want to express 7/x as (something) / (x+7). to do this, we can multiply 7/x  by 1.

anything divided by itself = 1. thus, if we multiply both the numerator and the denominator by something that turns x into (x+7), we can do what we want to do.

(x+7)/x * x turns x into (x+7), so we multiply both the numerator and denominator by (x+7)/x to get

$\frac{7}{x} = \frac{7(x+7)/x}{x+7}$

substitute this for 7/x in our original problem

$\frac{5x}{x+7} +\frac{7(x+7)/x}{x+7} = \frac{5x}{x+7} +\frac{(7x+49)/x}{x+7} = \frac{5x}{x+7} +\frac{7+49/x}{x+7} = \frac{5x+7+49/x}{x+7}$