Equations

Solving linear equations

A linear equation has the unknown only to the first power, like $3x - 7 = 11$ . To solve it, undo what was done to $x$ — doing exactly the same thing to both sides each time.

What is a linear equation?

An equation is a statement that two expressions are equal. In a linear equation the variable appears only as $x$ — never $x^2$ or $\frac{1}{x}$ . Solving means finding the value of $x$ that makes the statement true.

Think of a balance: both sides weigh the same. If you remove $5$ from one side, remove $5$ from the other and it stays balanced.

Try it: keep the scale level

Solve $2x + 6 = 14$ on a balance. Take cubes off one side and watch what happens — then fix it.

Worked examples

Two steps

Problem: $3x - 7 = 11$

Undo the subtraction: add $7$ to both sides.
$3x = 18$
Undo the multiplication: divide both sides by $3$ .
$x = 6$
Check in the original equation.
$3 \cdot 6 - 7 = 11 \;\checkmark$

Result: $x = 6$

Variable on both sides

Problem: $5x + 2 = 2x + 14$

Subtract $2x$ from both sides to gather the $x$ terms.
$3x + 2 = 14$
Subtract $2$ from both sides.
$3x = 12$
Divide by $3$ .
$x = 4$
Check: both sides equal $22$ .
$5 \cdot 4 + 2 = 22 = 2 \cdot 4 + 14$

Result: $x = 4$

With brackets

Problem: $2(x + 3) = 16$

Divide both sides by $2$ (or expand first — both work).
$x + 3 = 8$
Subtract $3$ .
$x = 5$

Result: $x = 5$

Common mistakes

Mistake 1 · Doing an operation to one side only.
Why it happens
The two sides stop being equal.
What to do instead
Write the operation under both sides every time.
Mistake 2 · Dividing only part of a side: $2x + 6 = 14 \to x + 6 = 7$ .
Why it happens
Dividing a side by $2$ means dividing every term on it.
What to do instead
Remove the $+6$ first, then divide: $2x = 8$ , $x = 4$ .
Mistake 3 · Using the wrong inverse: $2x + 6 = 14 \to 2x = 20$ .
Why it happens
The $+6$ is undone by subtracting $6$ , not adding it.
What to do instead
Ask “what was done to $x$ ?” and do the opposite.
Mistake 4 · Losing a sign when moving terms: $5x + 2 = 2x + 14 \to 7x = 12$ .
Why it happens
“Moving” $2x$ across actually means subtracting $2x$ from both sides.
What to do instead
Write the subtraction explicitly instead of “moving” terms.

Practice

1. Solve $4x + 5 = 29$ .
Hint
Subtract $5$ first.
Show answer
$4x = 24$ , so $x = 6$ .
2. Solve $7 - 2x = 1$ .
Hint
Subtract $7$ from both sides; watch the sign.
Show answer
$-2x = -6$ , so $x = 3$ .
3. Solve $3(x - 2) = 12$ .
Hint
Divide by $3$ first.
Show answer
$x - 2 = 4$ , so $x = 6$ .
4. Solve $6x - 4 = 2x + 8$ .
Hint
Gather the $x$ terms on one side.
Show answer
$4x = 12$ , so $x = 3$ .

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