Lesson Guide
Explanation
When x appears on both sides, the first job is to get all the x terms onto one side. In 7x − 4 = 3x + 16, subtract 3x from both sides to leave 4x − 4 = 16. Then it is an ordinary two-step equation: add 4 to get 4x = 20, and divide by 4 to get x = 5. There is a choice about which side to move, and a good habit is to move the smaller x term — subtracting 3x rather than 7x keeps the coefficient positive and avoids negatives, which is where most slips happen.
Your Turn
Solve 5x + 2 = 2x + 14. Check: subtract 2x → 3x + 2 = 14, so 3x = 12 and x = 4.
Watch Out
Whatever you subtract, subtract from BOTH sides. Removing 3x from only the right-hand side breaks the equation.
Explanation
A fraction with a whole expression on top is handled by multiplying it away. In (x + 2) over 3 equals 5, the entire bracket is being divided by 3, so multiply both sides by 3: x + 2 = 15, and then x = 13. The important idea is that a fraction bar acts like a bracket — it groups everything above it. So multiplying by 3 turns (x + 2) over 3 into x + 2, not into x + 6. Getting that wrong is the standard error here.
Your Turn
Solve (x − 4) ÷ 5 = 3. Check: multiply by 5 → x − 4 = 15, so x = 19.
Watch Out
Multiply the whole of the other side too. From (x + 2)/3 = 5, both sides get × 3 — the 5 becomes 15.
Explanation
When there are two or more fractions, clear them all at once by multiplying every term by the lowest common denominator. In x over 3 plus x over 4 equals 7, the denominators are 3 and 4, so the LCD is 12. Multiplying each term by 12: x over 3 becomes 4x, x over 4 becomes 3x, and the 7 becomes 84. That leaves 7x = 84, so x = 12. The trick is remembering to multiply the right-hand side as well — the number with no fraction is the one people forget.
Your Turn
Solve x/2 + x/5 = 7. Check: LCD is 10, giving 5x + 2x = 70, so 7x = 70 and x = 10.
Watch Out
Multiply EVERY term by the LCD, including any term that is not a fraction. Missing one is the usual mistake.
Explanation
When there is a single fraction on each side, cross-multiplying is quickest: multiply each numerator by the other side's denominator. From (2x − 1) over 5 equals (x + 4) over 4, cross-multiplying gives 4(2x − 1) = 5(x + 4). Expand both: 8x − 4 = 5x + 20. Subtract 5x and add 4 to get 3x = 24, so x = 8. Notice that cross-multiplying creates brackets automatically — the numerators must be treated as whole groups, which is exactly why they end up bracketed.
Your Turn
Solve (x + 1)/2 = (x − 3)/1. Check: x + 1 = 2(x − 3), so x + 1 = 2x − 6 and x = 7.
Watch Out
Cross-multiplying only works with ONE fraction on each side. With x/3 + x/4 = 7 you must use the LCD method instead.
Explanation
Four situations, four methods. If x is on both sides with no fractions, subtract the smaller x term. If there is one fraction against a plain number, multiply both sides by the denominator. If there are two or more fractions being added, multiply every term by the lowest common denominator. And if there is exactly one fraction on each side, cross-multiply. Whichever route you take, the aim is the same: after the first move there should be no fractions left at all. If any survive, you have missed a term.
Your Turn
Which method for (x + 5)/4 = (2x)/3? Check: one fraction each side → cross-multiply, giving 3(x + 5) = 8x.
Watch Out
Check your answer in the ORIGINAL equation with the fractions still in it — that is the only way to catch a term you failed to multiply.
📌 Key Points — this lesson in a nutshell
- x on both sides: gather the x terms first, moving the SMALLER one to keep the coefficient positive.
- A fraction bar acts like a bracket — it groups everything above it.
- One fraction: multiply both sides by its denominator. (x + 2)/3 = 5 → x + 2 = 15.
- Several fractions: multiply EVERY term by the lowest common denominator, including terms that are not fractions.
- One fraction each side: cross-multiply, and remember it creates brackets.
- After the first step no fractions should remain. If any do, a term was missed.
- Always check in the original equation, fractions included.
Step 1: Introduction
Step 1 of 5
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