Solving Simple Linear Equations

The big idea: keep the scale balanced

Picture an old-fashioned balance scale. The equals sign is the middle. The left side and the right side weigh exactly the same.

To find x, we want it sitting all alone on one side. The trick is to peel away everything around it using inverse operations, the "undo" buttons of math. And here is the golden rule:

Whatever you do to one side, you must do to the other side too.

That keeps the scale balanced, so the two sides stay equal the whole way through.

Meet the "undo" buttons

Every operation has an opposite that cancels it out. To move something away from x, use its inverse:

• Adding is undone by subtracting. (If something is added to x, subtract it from both sides.)
• Subtracting is undone by adding.
• Multiplying is undone by dividing. (If x is multiplied by a number, divide both sides by that number.)
• Dividing is undone by multiplying.

Worked example #1: x + 7 = 12 (one step)

The 7 is being added to x. To undo adding 7, we subtract 7 from both sides.

• Subtract: on the left, +7 and −7 cancel, leaving just x. On the right, 12 − 7 = 5.
• Result: x = 5.

Worked example #2: 3x = 15 (one step)

Here 3x means "3 times x." To undo multiplying by 3, we divide both sides by 3.

• Divide: on the left, 3 ÷ 3 = 1, so we are left with x. On the right, 15 ÷ 3 = 5.
• Result: x = 5.

The division partner, like x ÷ 4 = 2 (often written as x/4 = 2), works the same way in reverse: x is divided by 4, so you multiply both sides by 4, giving x = 2 × 4 = 8. Check: 8 ÷ 4 = 2. Correct.

Worked example #3: 2x + 3 = 11 (two steps)

When there are two things bundled around x, undo them one at a time. A handy rule of thumb: undo the adding and subtracting first, then the multiplying and dividing.

• Step 1, subtract 3 from both sides to clear the +3.
• Step 2, divide by 2 to undo the 2 multiplied by x.

Worked example #4: x + 5 + 4 = 16

Sometimes you will see two plain numbers on the same side as x. First, combine the numbers that are alone (5 + 4 = 9). Then it becomes a friendly one-step equation.

• Combine: 5 + 4 = 9, so the left side becomes x + 9.
• Subtract: take 9 off both sides. 16 − 9 = 7.
• Result: x = 7.

Tips that make this easier

• Always balance both sides. If you subtract 3 on the left, subtract 3 on the right. Treating only one side is the most common mistake.
• Undo plus and minus before times and divide. For two-step equations, clear the added or subtracted number first, then handle the multiply or divide.
• Always check your answer. Substitute it back into the original equation. If both sides match, you are done and you can move on with confidence.
• Write neatly, line by line. Keeping each step on its own row makes it easy to spot where a slip happened.

Your turn: practice problems

Solve each one by hand, then check your answer by substituting it back in. No peeking until you have tried.

1. x + 9 = 14
2. 5x = 35
3. x/3 = 6
4. 4x + 2 = 22