Question

how many solutions exist for the given equation?
12x + 1 = 3(4x + 1) - 2
zero two
infinitely many one

Explanation:

Step1: Expand the right side

First, we expand the right - hand side of the equation \(12x + 1=3(4x + 1)-2\). Using the distributive property \(a(b + c)=ab+ac\), where \(a = 3\), \(b = 4x\) and \(c = 1\), we get \(3(4x+1)=3\times4x+3\times1 = 12x+3\). So the equation becomes \(12x + 1=12x+3 - 2\).

Step2: Simplify the right side

Simplify the right - hand side of the equation. \(3-2 = 1\), so the equation is \(12x + 1=12x+1\).

Step3: Analyze the equation

Subtract \(12x\) from both sides of the equation \(12x + 1=12x+1\). We have \((12x + 1)-12x=(12x + 1)-12x\). Simplifying both sides, the left side is \(1\) and the right side is also \(1\). We get \(1 = 1\), which is a true statement for all values of \(x\). When we end up with a true statement (like \(a=a\) where \(a\) is a constant) after simplifying an equation, it means that the equation has infinitely many solutions.

Answer:

infinitely many