Linear equation: the 4-step method

Goal

Isolate $x$. Every linear equation reduces to $ax = b$, whose solution is $x = b/a$ (if $a \neq 0$).

4-step method

1. Expand if there are parentheses.
2. Gather: $x$-terms on the left, others on the right (flip sign when crossing the $=$).
3. Simplify on each side (collect like terms).
4. Divide by the coefficient of $x$.

Typical example

Solve $3(x - 2) = 5x + 4$.

1. Expand: $3x - 6 = 5x + 4$.

2. Gather: $3x - 5x = 4 + 6$.

3. Simplify: $-2x = 10$.

4. Divide: $x = \dfrac{10}{-2} = -5$.

Check: $3(-5 - 2) = -21$ and $5(-5) + 4 = -21$. ✓

Special cases

No solution: $0 x = c$ with $c \neq 0$. Ex: $2x + 3 = 2x - 1$ gives $0 = -4$, impossible.

Infinitely many solutions: $0 x = 0$. Ex: $2x + 3 = 2x + 3$ gives $0 = 0$, true for all $x$.

Equation with fractions

Multiply everything by the common denominator before solving.

Example: $\dfrac{x}{3} + 1 = \dfrac{x + 5}{6}$.

Multiply by $6$: $2x + 6 = x + 5$, then $x = -1$.

Common mistakes

1. Forgetting to flip signs when crossing the $=$.
2. Partial expansion (not distributing to every term).
3. Wrong division direction ($x = a/b$ vs $x = b/a$).
4. Not verifying by plugging back.

?À toi de jouer

Solve 2x + 7 = 4x − 3

A.x = 5B.x = −5C.x = 2D.x = 10

Valider