Question

solve the equation $3x + \frac{3}{5} = \frac{1}{3}(5x + 5)$

Explanation:

Step1: Simplify the right - hand side

We start with the equation \(3x+\frac{3}{5}=\frac{1}{3}(5x + 5)\). First, we distribute the \(\frac{1}{3}\) on the right - hand side. Using the distributive property \(a(b + c)=ab+ac\), where \(a=\frac{1}{3}\), \(b = 5x\) and \(c = 5\), we get \(\frac{1}{3}(5x+5)=\frac{5}{3}x+\frac{5}{3}\). So the equation becomes \(3x+\frac{3}{5}=\frac{5}{3}x+\frac{5}{3}\).

Step2: Eliminate fractions

To eliminate the fractions, we find the least common denominator (LCD) of 5 and 3, which is 15. Multiply each term in the equation by 15:
\(15\times(3x)+15\times\frac{3}{5}=15\times\frac{5}{3}x + 15\times\frac{5}{3}\)
Simplify each term:
\(45x + 9 = 25x+25\)

Step3: Move like terms together

Subtract \(25x\) from both sides:
\(45x-25x + 9=25x - 25x+25\)
\(20x+9 = 25\)
Then subtract 9 from both sides:
\(20x+9 - 9=25 - 9\)
\(20x=16\)

Step4: Solve for x

Divide both sides by 20:
\(x=\frac{16}{20}\)
Simplify the fraction by dividing both the numerator and the denominator by 4:
\(x=\frac{4}{5}\)

Answer:

\(x = \frac{4}{5}\)