Which of the following equations has no solution? | SAT Linear equations in one variable Question, Answer Explained

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Which of the following equations has no solution?

$3(x - 2) = 3x - 6$

$3(x - 2) = 2x - 6$

$3(x - 2) = 3x + 6$

$3(x - 2) = -3x + 6$

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Answer: C

The left side of each equation is $3x - 6$ . An equation of the form $3x - 6 = ax + b$ has no solution when the coefficients of $x$ match but the constants don't. In choice C, $3x - 6 = 3x + 6$ leads to $-6 = 6$ , which is never true, so the equation has no solution.

Choice A has the same coefficient of $x$ on both sides, but its constants also match, so it has infinitely many solutions, not none.

Why the other choices are wrong

A. Both sides equal 3x - 6, so every value of x is a solution. This equation has infinitely many solutions.
B. Subtracting 2x - 6 from both sides gives x = 0, so this equation has exactly one solution.
D. The x-coefficients are 3 and -3, so the x-terms don't cancel. Solving 6x = 12 gives the single solution x = 2.

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