Equations with x in the denominator

When x sits at the bottom

Equations like 3x + 2 = 14 you can already solve in your sleep. Today x hides in an unusual place: in the denominator. Like 6/x = 2.

The trick is simple: get rid of the fraction in the first step. Multiply both sides of the equation by the denominator. From 6/x = 2 you get 6 = 2x. And that is an equation you already know.

But watch out. You multiplied by an expression that has x in it. And if x happened to be zero, you would multiply by zero — and that breaks the equation. So here too a condition holds: the denominator must not be zero.

And that is why these equations need a check. You put the result back into the original question and check that the left side really equals the right. If the result came out as the forbidden number, there is no solution.

The method has three steps: condition, multiply, check. Come and live them.

Adam is training for entrance exams and gets stuck: “I got x = 1, but in the question there is 5/(x − 1). When I substitute, I divide by zero!” Ema nods: “That is exactly why you write the condition. Here it is x ≠ 1 — and if you get exactly one, it means the equation has no solution. That is not a mistake in the calculation. That is the right answer: no solution.” Adam writes it in big letters. The answer “no solution” is just as valid in entrance exams as any number.

💡

Find the condition before you start calculating. At the end you just glance: did the forbidden number come out? If yes, the equation has no solution.

Three steps: condition, multiply, check

Step 1: Condition. Find the denominator and say when it is zero. For the equation 12/(x − 1) = 4 the denominator is x − 1, so x ≠ 1.

Step 2: Multiply both sides by the denominator. Left side: the fraction disappears, the numerator stays. Right side: the number times the denominator. From 12/(x − 1) = 4 you get 12 = 4 · (x − 1).

Step 3: Finish it as an ordinary equation. Expand the bracket: 12 = 4x − 4. Add 4: 16 = 4x. Divide by four: x = 4.

Step 4: Compare with the condition. You got x = 4 and the forbidden one was one. Four passes.

Step 5: Check. Substitute into the original question: 12/(4 − 1) = 12/3 = 4. The right side is 4. Left = right, 完成.

The check here is not extra nagging — it is part of the solution. By multiplying by the denominator, a number can sneak into the calculation that does not work in the original equation. The check spots it.

Example 1: 6/x = 2

Question: Solve the equation 6/x = 2.

Method: Condition: x ≠ 0.

Multiply both sides by x: 6 = 2x.

Divide by two: x = 3.

Compare with the condition: 3 is not the forbidden number, fine.

Check: Left side: 6/3 = 2. Right side: 2. Equal.

Answer: x = 3.

Example 2: 15/(x + 2) = 3

Question: Solve the equation 15/(x + 2) = 3.

Method: Condition: x + 2 ≠ 0, so x ≠ −2.

Multiply both sides by the expression (x + 2): 15 = 3 · (x + 2).

Expand: 15 = 3x + 6.

Subtract 6: 9 = 3x. Divide by three: x = 3.

Check: Left side: 15/(3 + 2) = 15/5 = 3. Right side: 3. It matches.

Answer: x = 3.

Example 3: an equation with no solution

Question: Solve the equation (2x − 6)/(x − 3) = 5.

Method: Condition: x ≠ 3.

Multiply: 2x − 6 = 5 · (x − 3), so 2x − 6 = 5x − 15.

Move: −6 + 15 = 5x − 2x, so 9 = 3x, so x = 3.

Compare with the condition: Three is forbidden! You cannot substitute it — the denominator would be zero.

Answer: The equation has no solution. And that is a full answer, not giving up.

⚠️

The check is not an optional bonus. With equations with x in the denominator, the “result” can come out as the forbidden number — and then there is no solution. Without a check (or at least a comparison with the condition) you hand in an answer that does not work in the original equation at all. Thirty seconds of checking will save the whole question.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. 开始 with ten. When it goes well, add more.

Warm up your hand on ordinary equations — you will turn every equation with a fraction into one of those. Write whole numbers.

Practise the questions

On paper

Three equations, each time the whole ritual: condition, multiply, finish, check.

Solve 8/x = 4. Write the condition, multiply both sides by x, finish and do the check.
Solve 10/(x − 2) = 5. Watch the bracket when you multiply.
Solve 9/(x + 1) = 3 and do the check by substituting into the original question.
For every equation check: did the forbidden number from the condition come out?

Now you 💪

You can explain why both sides are multiplied by the denominator and what that does to the fraction.
Every equation on paper has a condition and a check.
You know what it means when the result comes out as the forbidden number — the equation has no solution.

✅

Done when: You have three solved equations with x in the denominator, a condition and a check for each, and all the checks match.

What to take from this lesson

You get rid of a fraction in an equation by multiplying both sides by the denominator.
The condition (denominator ≠ 0) is written before you calculate.
After solving, compare the result with the condition.
A check by substituting into the original question is a required part of the solution.
“No solution” is a valid answer when the forbidden number comes out.