Equations with negative numbers
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Solve −2x + 5 = 13.
Common mix-up
In this example, a student wrote −2x = 8, so x = 4.
Subtracting 5 was correct. Dividing by positive 2 lost the negative sign attached to the multiplier.
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See why this works
Where did the negative sign go?
A negative sign belongs to the number next to it. Keeping that number together can prevent an almost-correct equation from turning into the wrong answer.
In −2x, the multiplier of x is −2. Divide by −2 to undo it.
Start with −2x + 5 = 13. The term −2x means negative two multiplied by x. Think of −2 as one signed number, rather than a 2 with a decoration that can disappear. This matters when you choose how to undo the multiplication.
Subtract 5 from both sides. You get −2x = 8. This step is correct: 13 − 5 is 8, and the negative sign remains attached to the 2. The mistake would be dividing by positive 2 and writing x = 4. The multiplier you need to undo is −2.
Divide both sides by −2 instead. The left side becomes x because −2 divided by −2 is 1. The right side is 8 divided by −2, which is −4. Therefore x = −4. A positive number divided by a negative number gives a negative result.
Check the original equation using parentheses around the negative answer: −2(−4) + 5 = 8 + 5 = 13. The two negative factors make a positive product. If you tried positive 4, the left side would be −8 + 5 = −3, which does not equal 13. Substitution helps you catch a missing sign without having to repeat every solving step.
Keep this idea
Carry each sign with its number. Undo multiplication with the signed multiplier, and substitute your answer into the original equation.
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