Linear Equations in One Unknown · Little Blue Book, Visualized

What this move does

An equation is like a balance that is level. It has two properties:

A = B ⟹ A \pm c = B \pm c, A \times c = B \times c, \frac{A}{c} = \frac{B}{c} (c \neq = 0)

In words: add the same number to both sides, subtract it, multiply by it, or divide by the same number as long as it is not 0, and the balance stays level. Solving a linear equation in one unknown means using these two properties again and again, step by step, until $x$ is alone on one side:

Clear the denominators first, then remove the brackets.
Move terms: put the terms with $x$ on one side and the numbers on the other.
Collect like terms, then make the coefficient of $x$ equal to $1$ .
Substitute the number you get back into the original equation to check.

The balance above only lets you “change both sides together”, so however you click, it stays level. Some routes just take longer than others.

Predict first

Do not spend long on paper: what is the solution of $5 x - 4 = 2 x + 8$ ?

Choose an answer first; then you can see the result.

Work through one

Example: solve 3x + 5 = x − 7

Step 1 / 5

Look at both sides
The left pan holds $3$ boxes and $5$ weights; the right pan holds $1$ box and $7$ “negative weights”. Both sides have $x$ , so first get $x$ onto one side only.

$3 x + 5 = x - 7$
Subtract x from both sides
Subtract 5 from both sides
Divide both sides by 2
Substitute back to check

What about denominators?

Fraction coefficients are easy to get wrong. First multiply both sides by the least common multiple of all the denominators, and they vanish in one go. In the equation below the denominators are $2$ and $3$ , so start by multiplying by $6$ . Before you do, think: there are four things on the balance. What does each one become?

Solve

\frac{x}{2} + 1 = \frac{x}{3} + 2

. For the first step, click “× 6”.

When it fails

Break it

Counterexample lab: remove one condition

Conclusion: the balance stays level and the solution does not change

Here you can only change both sides together. Click whatever you like: the balance never tips.

Practice

Exercise 1Basic

Solve $4 x - 7 = 13$ .

Exercise 2Basic

Solve $7 - 2 x = 3 x - 8$ .

Exercise 3Advanced

Solve $2 (x - 3) - 3 (1 - x) = x + 7$ .

Exercise 4Challenge

Solve $\frac{x + 1}{2} - \frac{2 x - 1}{3} = 1$ .

Mistake Museum · Exhibits from this stop

Before you leave, look at the traps other people fell into here.See every exhibit in the museum →

Exhibit

Moving a term without changing its sign

3 x + 5 = x - 7

; moving terms gives

3 x + x = - 7 + 5

What went wrong? Think first, then open

Diagnosis “Moving $x$ from the right to the left” really means subtracting $x$ from both sides, so what appears on the left is $- x$ , not $+ x$ . The same goes for moving $5$ to the right: you subtract $5$ from both sides.

3 x - x = - 7 - 5

, so

2 x = - 12

and

x = - 6

. Check: left side

3 \times (- 6) + 5 = - 13

, right side

- 6 - 7 = - 13

“Moving a term means adding or subtracting on both sides; crossing the equals sign flips its sign”

Exhibit

Clearing denominators but missing terms without one

\frac{x}{2} + 1 = \frac{x}{3} + 2

; multiply both sides by

6

to get

3 x + 1 = 2 x + 2

, so

x = 1

What went wrong? Think first, then open

Diagnosis Multiplying both sides by $6$ means multiplying every term by $6$ . The $1$ and the $2$ have no denominator, but they are on the balance too, so they must become $6$ and $12$ . Substituting back gives it away: the left side is $\frac{3}{2}$ , the right side is $\frac{7}{3}$ .

3 x + 6 = 2 x + 12

, so

x = 6

. Check: left side

3 + 1 = 4

, right side

2 + 2 = 4

“When you multiply, everything on the balance gets multiplied”