How do you solve an equation with the unknown on both sides?

Gathering the unknown

When x appears on both sides, it cannot be isolated yet. The first step is to collect all the x terms on one side and all the numbers on the other.

After gathering, the equation looks exactly like the ones in the previous lesson and is solved the same way. There is no new method here, only one preparatory step.

Why the sign flips

Moving a term across sounds like an arbitrary rule, and it is not. Moving two x to the right means subtracting two x from both sides — and on the right that is exactly what flips the sign.

So there is nothing to memorise. Anyone who writes the subtraction out once sees the sign flip by itself, and from then on moving is a shortcut rather than a rule.

It pays to gather the x on whichever side has the larger coefficient. That keeps it positive and saves a line of working with a minus.

Two special cases

Sometimes the x terms cancel entirely and a false statement is left, such as three equals five. That means no value of x satisfies the equation — it has no solution.

And sometimes what is left is always true, such as five equals five. Then every number solves it, and it is really an identity rather than an equation.

Worked examples

1. Solve: 5x plus 2 equals 3x plus 10

   1. Move 3x left: 5x minus 3x gives 2x
   2. Move 2 right: 10 minus 2 gives 8
   3. This gives 2x equals 8, so divide by 2

   Answer: x equals 4. Check: both sides come out 22

2. Solve: 2x plus 6 equals 2x plus 1

   1. Move 2x left: the x terms cancel completely
   2. What remains is 6 equals 1
   3. That is false for every value of x

   Answer: No solution

3. Solve: 4x minus 3 equals 2(2x − 1) minus 1

   1. Expand on the right: 4x minus 2, then minus 1
   2. Collect: 4x minus 3
   3. Both sides are identical, so every x satisfies it

   Answer: Every number is a solution

Common mistakes

Moving a term without flipping its sign

Moving is subtracting from both sides, so the sign must flip. Writing the subtraction out shows it, and substituting back catches it.

Gathering the x on the side with the smaller coefficient

It is correct but leads to a negative coefficient and an extra line of working. Choosing the side with the larger coefficient keeps it positive.

Writing zero when the x terms cancel

If a false statement is left, the answer is that there is no solution, not that x equals zero. Zero is a number, and in that case it does not solve it either.

More in Linear Equations

• How do you turn a sentence into an algebraic expression?
• How do you collect like terms and expand brackets?
• How do you solve a first degree equation?
• How do you solve an equation with brackets and with fractions?
• How do you turn a word problem into an equation?