Question

1. \\(1 + 7(5x - 1) = 6(6x - 1) - x\\)

beige \t green \t purple
\t \\(x = 0\\) \t \\(\infty\\)

Explanation:

Step1: Expand both sides

Expand \(7(5x - 1)\) and \(6(6x - 1)\) using the distributive property \(a(b + c)=ab+ac\).
Left side: \(1+7(5x - 1)=1 + 35x-7=35x - 6\)
Right side: \(6(6x - 1)-x = 36x-6 - x=35x - 6\)

Step2: Analyze the equation

We have the equation \(35x - 6=35x - 6\).
Subtract \(35x\) from both sides: \(- 6=-6\)
This is a true statement for all real numbers \(x\), meaning the equation has infinitely many solutions. But looking at the given option \(x = 0\), let's check if \(x = 0\) satisfies the original equation.
Substitute \(x = 0\) into left side: \(1+7(0 - 1)=1-7=-6\)
Substitute \(x = 0\) into right side: \(6(0 - 1)-0=-6 - 0=-6\)
Left side equals right side when \(x = 0\), and since the equation is an identity (both sides are equal for all \(x\)), \(x = 0\) is a solution (among infinitely many).

Answer:

\(x = 0\) (the solution \(x = 0\) is valid as it satisfies the equation, and the equation is an identity so all real numbers are solutions, but \(x = 0\) is one of them)