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QuestionSolve the equation: 75x+10=7x+2−45\frac{7}{5 x+10}=\frac{7}{x+2}-\frac{4}{5}

Studdy Solution

STEP 1

Assumptions1. We are given the equation 75x+10=7x+−45\frac{7}{5x+10}=\frac{7}{x+}-\frac{4}{5}. . We are tasked to find the value of xx that satisfies this equation.

STEP 2

First, we can simplify the equation by multiplying every term by the least common multiple (LCM) of the denominators to eliminate the fractions. The LCM of 5x+105x+10, x+2x+2, and 55 is 5(x+2)5(x+2).
5(x+2)(75x+10)=5(x+2)(7x+2)−5(x+2)(45)5(x+2)\left(\frac{7}{5x+10}\right)=5(x+2)\left(\frac{7}{x+2}\right)-5(x+2)\left(\frac{4}{5}\right)

STEP 3

implify the equation by cancelling out the common factors.
7(x+2)=35−(5x+10)7(x+2)=35-(5x+10)

STEP 4

Expand the brackets on both sides of the equation.
7x+14=35−20x−407x+14=35-20x-40

STEP 5

Rearrange the equation to bring all terms involving xx to one side and constants to the other side.
7x+20x=35−14−407x+20x=35-14-40

STEP 6

Combine like terms on both sides of the equation.
27x=−1927x=-19

STEP 7

Finally, solve for xx by dividing both sides of the equation by 2727.
x=−1927x=\frac{-19}{27}So, the solution to the given equation is x=−1927x=\frac{-19}{27}.

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