Question

which expression is equivalent to \\(\frac{2}{3}x + 1\\)?
a \\(\frac{1}{3}(2x + 3)\\)
b \\(\frac{2}{3}(2x + 3)\\)
c \\(\frac{2}{3}(3x + 2)\\)
d \\(\frac{2}{3}(3x + 1)\\)

Explanation:

Step1: Expand Option A

Expand \(\frac{1}{3}(2x + 3)\) using the distributive property \(a(b + c)=ab+ac\). Here \(a=\frac{1}{3}\), \(b = 2x\), \(c = 3\). So \(\frac{1}{3}(2x+3)=\frac{1}{3}\times2x+\frac{1}{3}\times3=\frac{2}{3}x + 1\).

Step2: Check Other Options (Optional)

• For Option B: \(\frac{2}{3}(2x + 3)=\frac{4}{3}x+2\), not equal to \(\frac{2}{3}x + 1\).
• For Option C: \(\frac{2}{3}(3x + 2)=2x+\frac{4}{3}\), not equal to \(\frac{2}{3}x + 1\).
• For Option D: \(\frac{2}{3}(3x + 1)=2x+\frac{2}{3}\), not equal to \(\frac{2}{3}x + 1\).

Answer:

A. \(\frac{1}{3}(2x + 3)\)