Linear Equations • Topic 2 of 4

Isolating Variables

Isolating a variable means peeling away everything attached to it, one operation at a time, using inverse operations in reverse of the order of operations. If x has been multiplied then had a number added, you undo the addition first and the multiplication second. Keep the equation balanced at each step. When the variable appears on both sides, collect it on the side where its coefficient stays positive to avoid sign errors. When a variable sits in a denominator, multiply both sides by that denominator to lift it out first. A useful GRE shortcut: you often do not need the value of x itself — if a question asks for 3x + 1 and you can find 3x directly, stop there rather than solving fully for x.

✅ Solved examples

1. Solve (x/4) + 3 = 10.
Subtract 3: x/4 = 7. Multiply by 4: x = 28.
2. Solve 12 = 5 − 2x.
Subtract 5: 7 = −2x. Divide by −2: x = −7/2.
3. If 3x + 5 = 20, find the value of 6x + 1 without fully isolating x.
3x = 15, so 6x = 30 and 6x + 1 = 31 (no need to divide to x = 5).
4. Solve 8/x = 2 for x.
Multiply both sides by x: 8 = 2x, so x = 4.

✏️ Practice — try these, take hints as needed

1. Solve (x/3) − 2 = 4.
Add 2.
x/3 = 6.
Multiply by 3.
x = 18
2. Solve 9 = 4 − x.
Subtract 4.
5 = −x.
Multiply by −1.
x = −5
3. If 2x − 7 = 5, find 4x + 3.
2x = 12.
4x = 24.
Add 3.
27
4. Solve 15/x = 3.
Multiply both sides by x.
15 = 3x.
Divide by 3.
x = 5
5. Solve 6 − (x/2) = 1.
Subtract 6.
−x/2 = −5.
Multiply by −2.
x = 10

📝 Topic test — 8 questions

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