A Contradiction in a Linear Equation
Leo is solving a linear equation on the Mathematics Achievement section: 3(x - 4) = 5x + 2 - 2(x + 3). He distributes the left side to get 3x - 12. On the right, he distributes the -2 as -2x - 6, so the right side becomes 5x + 2 - 2x - 6, which simplifies to 3x - 4. The equation is now 3x - 12 = 3x - 4. He subtracts 3x from both sides and gets -12 = -4. Leo pauses. He has seen equations with 'no solution' before, but he also wonders whether he made a sign error when distributing the -2. He checks by substituting x = 0 into the original equation: left side = 3(0 - 4) = -12; right side = 5(0) + 2 - 2(0 + 3) = 2 - 6 = -4. The two sides are not equal. He must decide whether the equation has no solution, one solution, or infinitely many solutions, and explain how the algebraic result supports that conclusion.
1. Verify Leo's distribution and simplification. Is his work correct? If so, what does -12 = -4 mean about the equation?
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Leo's distribution is correct. On the left, 3(x - 4) = 3x - 12. On the right, 5x + 2 - 2(x + 3) = 5x + 2 - 2x - 6 = 3x - 4. The equation becomes 3x - 12 = 3x - 4. Subtracting 3x from both sides gives -12 = -4. That statement is false and contains no variable. Because the variable cancels and the resulting equation is false, no value of x can satisfy the original equation. The equation has no solution. This is different from an identity, such as -12 = -12, which would mean every value of x works.
2. Leo's substitution of x = 0 gives different values on each side. Does that alone prove there is no solution? Explain what it does and does not show.
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Substituting x = 0 shows that 0 is not a solution: the left side is -12 and the right side is -4. That is useful as a check, but it does not prove there is no solution, because some other value of x might still work. A single failed test eliminates only that one value. The algebraic contradiction -12 = -4 is stronger because it holds after the variable has been eliminated, so it applies to every possible x. Leo's substitution confirms that his simplified equation is not accidentally true at x = 0, but the proof of 'no solution' comes from the contradiction, not from the single test.
3. Describe a common distribution error that would lead to a different conclusion, and show the incorrect result it would produce.
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A common error is to distribute 3(x - 4) as 3x - 4, forgetting to multiply -4 by 3. With that mistake, the equation becomes 3x - 4 = 3x - 4. Subtracting 3x gives -4 = -4, which is true for every x, so Leo would conclude there are infinitely many solutions. The correct result is no solution. This error is dangerous because substituting x = 0 into the mistaken equation gives -4 = -4, which looks consistent; the error is exposed by rechecking the distribution itself, not by testing a single value. Leo's habit of writing out each distribution step helps prevent this.