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Level 9Grade 9skill: linear_equations· 9 min read

Solving Linear Equations

In short: A linear equation stays true only if you do the same thing to both sides. "Moving a term across the equals sign" is just subtracting it from both sides — which is exactly why its sign flips. Clear brackets and fractions first, gather the variable on one side, reduce to ax = b, then divide b by a (never a by b), and substitute back to check.

How to solve a linear equation

A linear equation is a balance. The left side weighs exactly the same as the right side, and your only job is to keep that balance true while you strip everything away from x until it stands alone. Whatever you do — add, subtract, multiply, divide — you do to both sides at once. That single rule is the whole game.

You will hear teachers say "move the term to the other side and flip its sign." That is a shortcut, not a new law. Moving +6 across the equals sign as −6 is just a fast way of writing "subtract 6 from both sides." Understanding that this is why the sign flips will save you from the most common mistake in the topic.

Work in a fixed order so you never have to guess what to do next. Here is the sequence to follow every time:

  1. Expand every bracket and clear fractions by multiplying each term by the lowest common denominator.
  2. Collect all the variable terms on one side and all the plain numbers on the other. Every term that crosses the equals sign flips its sign.
  3. Combine like terms until you have a single form: ax = b.
  4. Divide both sides by a. Keep the order b divided by a, never a divided by b.
  5. Substitute your answer back into the original equation to confirm both sides match.

Why terms flip sign when they cross the equals sign Why it works

Term crosses =
Adding or subtracting the same term on both sides looks like the term jumping the equals sign with its sign reversed. Nothing magical — it is one subtraction written quickly.

x + 6 = 10 means x = 10 − 6 = 4

Factor crosses =
A number multiplying the whole side is undone by dividing both sides; a number dividing the side is undone by multiplying. Direction matters, so keep b ÷ a.

3x = 21 means x = 21 ÷ 3 = 7

Collect the variable
Drag the x-terms to whichever side keeps the coefficient positive. Chasing a negative coefficient is where careful students still drop a sign.

4 − 3x = 2x + 19 becomes 4 − 19 = 5x

Here is the one insight worth carrying: the equals sign is a mirror, not a doorway. A term does not 'travel' anywhere — you are removing it from one side, and the balance forces you to remove it from the other too. When you truly see that, the sign flip stops being a rule to memorise and becomes something you could re-derive on the spot, even half-asleep in an exam.

Worked examples

Solve 5x − 7 = 3x + 11. = x = 9
  1. Subtract 3x from both sides to gather the variable on the left: 2x − 7 = 11.
  2. Add 7 to both sides: 2x = 18.
  3. Divide both sides by 2: x = 9.
  4. Check: 5(9) − 7 = 38 and 3(9) + 11 = 38. Balanced.
Solve 4 − 3x = 2x + 19. = x = −3
  1. Move the x-terms to the side that keeps them positive. Add 3x to both sides: 4 = 5x + 19.
  2. Subtract 19 from both sides: −15 = 5x.
  3. Divide both sides by 5: x = −3.
  4. Check: 4 − 3(−3) = 13 and 2(−3) + 19 = 13. Balanced. A student who wrote 2x − 3x = −x and mishandled the sign would land on the wrong answer here — the negative coefficient is the trap.
Solve (2x + 1)/3 − (x − 2)/4 = 1. = x = 2/5
  1. Multiply every term by the lowest common denominator, 12: 4(2x + 1) − 3(x − 2) = 12.
  2. Expand carefully. The minus sign in front of 3 hits both terms: 8x + 4 − 3x + 6 = 12.
  3. Combine like terms: 5x + 10 = 12.
  4. Subtract 10: 5x = 2, so x = 2/5.
Solve 3(2x − 4) − 2(x − 5) = 2(x + 1). = x = 2
  1. Expand each bracket, watching the −2: 6x − 12 − 2x + 10 = 2x + 2.
  2. Simplify the left side: 4x − 2 = 2x + 2.
  3. Subtract 2x and add 2 to both sides: 2x = 4.
  4. Divide by 2: x = 2. (Miss the sign on −2(x − 5) and you get a tempting but wrong x = 12.)
Emma buys 12 items in total — pens at $3 each and notebooks at $5 each — and spends $46. How many pens did she buy? = 7 pens
  1. Let p be the number of pens, so notebooks number 12 − p.
  2. Write the cost equation: 3p + 5(12 − p) = 46.
  3. Expand: 3p + 60 − 5p = 46, which simplifies to −2p + 60 = 46.
  4. Subtract 60: −2p = −14, so p = 7. Emma bought 7 pens and 5 notebooks. Check: 3(7) + 5(5) = 21 + 25 = 46.

Common mistakes

Forgetting to flip the sign when a term crosses over

✗ For 4x + 9 = x + 30, writing 3x = 30 + 9 = 39, so x = 13.

After subtracting x from both sides you have 3x + 9 = 30. The +9 must leave the left side, which means subtracting 9 from both sides — it arrives on the right as −9, not +9. Adding instead of subtracting inflates the answer.

Fix: 3x + 9 = 30 gives 3x = 30 − 9 = 21, so x = 7. Say it out loud: a term that crosses the equals sign changes sign. Check by substituting: 4(7) + 9 = 37 and 7 + 30 = 37.

sign_error

Isolating the term but not finishing the division

✗ For 5x + 4 = 39, reaching 5x = 35 and writing x = 35.

You have isolated the term 5x, but 5x is not x. The variable is still multiplied by 5, so one operation remains. Stopping here reports the value of five x's, not one.

Fix: Divide both sides by 5: x = 35 ÷ 5 = 7. Whenever your last line still has a number stuck to the x, you are not done.

partial_computation

Dividing the wrong way at the final step

✗ For 9x = 3(x + 4), which reduces to 6x = 12, writing x = 6 ÷ 12 = 1/2.

To undo 'six times x', you divide the other side by 6, giving 12 ÷ 6. Flipping it to 6 ÷ 12 divides in the wrong direction and turns a whole number into a fraction that will not check out.

Fix: ax = b always gives x = b ÷ a. So x = 12 ÷ 6 = 2. Confirm: 9(2) = 18 and 3(2 + 4) = 18.

reversed_operands

Undoing operations in the wrong order

✗ For x/3 − 5 = 1, multiplying by 3 first and writing x − 5 = 3, so x = 8.

The x has two things done to it: divided by 3, then 5 subtracted. To reverse, you undo the last operation first — add 5 — before multiplying by 3. Jumping straight to multiplying by 3 mishandles the −5 term and skips a step.

Fix: Add 5 first: x/3 = 6. Then multiply both sides by 3: x = 18. Undo operations in reverse order, like taking off shoes before socks. Check: 18/3 − 5 = 1.

order_of_ops_lr

Practice

Solve a mixed set of linear equations — variables on both sides, brackets, fractions, and word problems. Take each one all the way to a fully checked answer.

Related skills

First, make sure of

Frequently asked questions

Why does a term change sign when it moves to the other side?

Because it is not really moving. "Moving" +6 across the equals sign is shorthand for subtracting 6 from both sides. The 6 disappears from one side and shows up on the other as −6. The sign flip is the visible result of doing the same operation to both sides, not a separate rule.

Should I subtract or divide first?

Subtract or add first, divide last. Reverse the order the operations were applied to x: whatever was done last is undone first. For 3x + 5 = 20, the +5 was applied last, so remove it first (3x = 15), then divide by 3 to get x = 5.

What if x ends up on the right, or the coefficient is negative?

Both are fine. x = 7 and 7 = x mean the same thing. To avoid sign slips, collect the variable on whichever side keeps its coefficient positive. If you do end with a negative coefficient like −2x = −14, just divide by −2 to get x = 7.

How do I know my answer is correct?

Substitute it back into the original equation and check both sides give the same number. If the left and right match, you are right; if they do not, you have almost certainly dropped a sign or divided the wrong way at the last step.