Question

to remove the fractions in an equation, both sides can be multiplied by the least common denominator. what is the least common denominator of the fractions in the following equation?
\\(\frac{5}{3}x - \frac{1}{9} = \frac{2}{9}x + \frac{17}{3}\\)
fill in the blanks to show what the equation will be once both sides are multiplied by the least common denominator.
\\(\square x - 1 = 2x + \square\\)

Explanation:

Step1: Identify denominators

The fractions in the equation \(\frac{5}{3}x - \frac{1}{9}=\frac{2}{9}x+\frac{17}{3}\) have denominators \(3\) and \(9\).

Step2: Find LCD of 3 and 9

The least common denominator (LCD) of \(3\) and \(9\) is \(9\) (since \(9\) is a multiple of \(3\)).

Step3: Multiply each term by 9

• For \(\frac{5}{3}x\): \(9\times\frac{5}{3}x = 15x\) (because \(9\div3 = 3\), \(3\times5 = 15\))
• For \(-\frac{1}{9}\): \(9\times(-\frac{1}{9})=- 1\)
• For \(\frac{2}{9}x\): \(9\times\frac{2}{9}x = 2x\)
• For \(\frac{17}{3}\): \(9\times\frac{17}{3}=51\) (because \(9\div3 = 3\), \(3\times17 = 51\))

Answer:

The least common denominator is \(9\). The equation after multiplying by \(9\) is \(15x - 1=2x + 51\). So the blanks are filled with \(15\) and \(51\) respectively.