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Day 1 · Learn3 min read
Solving One-Variable Linear Equations
An equation is a balance. The left side equals the right side, and your job is to find the value of the variable that keeps them equal. Solving means getting the variable alone on one side, like x = a number.
The golden rule
Whatever you do to one side, you must do to the other. This keeps the balance even. To undo an operation, use its opposite: addition undoes subtraction, and multiplication undoes division.
If a number is added, subtract it from both sides.
If a number is subtracted, add it to both sides.
If the variable is multiplied by a number, divide both sides by that number.
The plan
Peel away the numbers around the variable in reverse order. Undo any addition or subtraction first, then undo the multiplication or division last.
3x + 5 = 203x = 15 → x = 5
Example. Solve 2x − 7 = 9. Add 7 to both sides: 2x = 16. Divide both sides by 2: x = 8. Check: 2(8) − 7 = 16 − 7 = 9. ✓ It works, so x = 8 is correct.
Always check your answer by putting it back into the original equation. If both sides come out equal, you solved it right.
Now work today's 5 problems ↓
Q1
Solve for x: 2x + 6 = 20
Undo the addition first: subtract 6 from both sides to get 2x = 14. Then undo the multiplication: divide both sides by 2, giving x = 7. Dividing before subtracting the 6, or forgetting to divide by 2, leads to a wrong value.
Q2
Solve for x: x/3 − 4 = 2
Add 4 to both sides to isolate the fraction: x/3 = 6. Then multiply both sides by 3 to undo the division, giving x = 18. Subtracting 4 instead of adding it flips the sign, and skipping the multiply-by-3 step leaves the answer too small.
Q3
Solve for x: 7x − 4 = 3x + 20
Move the variable terms to one side by subtracting 3x from both sides: 4x − 4 = 20. Then add 4 to both sides: 4x = 24. Divide by 4 to get x = 6. Subtracting the larger 7x instead (making −4x) reverses the sign, and forgetting the final division leaves 24.
Q4
Solve for x: 2(x + 4) = 22
Distribute the 2 to both terms inside the parentheses: 2x + 8 = 22. Subtract 8 from both sides: 2x = 14. Divide by 2 to get x = 7. Multiplying only the x (leaving +4) or adding 8 instead of subtracting it produces the wrong value.
Q5
Solve for x: (2x − 1)/3 + 4 = x
Multiply every term by 3 to clear the denominator: 2x − 1 + 12 = 3x, which simplifies to 2x + 11 = 3x. Subtract 2x from both sides to get 11 = x. Forgetting to multiply the 4 by 3, or mishandling the sign of the −1, gives one of the other values.
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