Linear equations
Equations like ax + b = c, with brackets and with fractions, and the special cases with no solution or infinitely many.
How to solve it
An equation is a balance: whatever you do to both sides keeps it balanced. The goal is to leave $x$ alone on one side. Terms move across with the opposite sign, and you divide by the coefficient of $x$.
Step by step
- If there are fractions, multiply both sides by the common denominator.
- Expand the brackets.
- Move the terms with $x$ to the left and the numbers to the right, changing the sign as they cross.
- Collect like terms and divide both sides by the coefficient of $x$.
- If the coefficient of $x$ becomes zero: $0 = 0$ means infinitely many solutions, $0 = 5$ means none.
- Check by substituting the root into the original equation.
Common mistakes
- Moving a term without changing its sign: $5x - 4 = 16$ gives $5x = 20$, not $5x = 12$.
- Expanding $-6(x + 3)$ and changing only the first sign: it is $-6x - 18$.
- Not multiplying every term by the common denominator: in $\frac{x}{2} + \frac{x}{6} = 12$ the right side is multiplied by $6$ too, $3x + x = 72$.
- Seeing $0 = 0$ and answering “$x = 0$”. In fact every $x$ works.