Unit 4 · Topic 01 · Linear Equations
Kabir solved 2x + 3 = 11 by writing x = 11 + 3 − 2, and got a number that made no sense when he checked it.
Zara looked at his working and said he'd moved a number across the equals sign without changing its job.
One flipped sign later, the equation actually balanced.
Kabir was solving 2x + 3 = 11 for his homework. He wrote x = 11 + 3 − 2, mixing up which operations to undo and in what order, and got x = 12, which didn't check out when he substituted back: 2(12)+3=27, not 11. Zara showed him the real method: an equation is a balance — whatever you do to one side, you must do to the OTHER side too, to keep it balanced.
"Start by isolating the term with x," she said. "Subtract 3 from BOTH sides: 2x + 3 − 3 = 11 − 3, giving 2x = 8. Then divide BOTH sides by 2: x = 4." Kabir checked: 2(4)+3=8+3=11. It matched.
"So the rule is," Kabir said, "whatever moves across the equals sign, its operation flips — plus becomes minus, times becomes divide." Zara nodded but added a caution: "That's a useful shortcut once you understand WHY — it's really just doing the same operation to both sides and having the terms cancel. Never apply the shortcut blindly without understanding the balance behind it."
She then gave him a harder one: 3x − 5 = 2x + 7. "Get all the x terms on one side, and all the numbers on the other," she said. Subtract 2x from both sides: x − 5 = 7. Add 5 to both sides: x = 12. Kabir checked: 3(12)−5=31, and 2(12)+7=31. Both sides matched.
Kabir then tried a word problem: "A number, when 7 is added to twice it, gives 25. Find the number." He translated it into an equation: 2x + 7 = 25. Subtracting 7: 2x = 18. Dividing by 2: x = 9.
By the end, Kabir had a clear method for any linear equation: simplify each side first if needed, move all variable terms to one side and all constants to the other (using equal operations on both sides), then divide to isolate the variable — and always check the answer by substituting it back into the ORIGINAL equation.
A linear equation in one variable has the variable appearing only to the first power (no x², no 1/x). Solving it means finding the value that makes both sides equal.
The core rule: whatever operation you perform, do it to BOTH sides of the equation, to keep it balanced. This is what justifies the 'move across, flip the sign/operation' shortcut.
To solve: collect all variable terms on one side and all constant terms on the other side, using equal operations on both sides, then divide to isolate the variable.
Always verify a solution by substituting it back into the ORIGINAL equation — both sides must come out equal.
Solve 5x − 4 = 3x + 10 by collecting variable terms on one side and constants on the other, then check your answer.
Translate this into an equation and solve it: 'A number, when 9 is subtracted from three times it, gives 21.'
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