Linear Equations

A linear equation is simply an equation where $x$ only appears to the first power, with no $x^2$, no $\sqrt x$, nothing fancier. It's called "linear" because its graph is a straight line. The formula up top shows the whole idea: in $ax + b = c$, $x$ got multiplied by $a$ and then had $b$ added, so to get it back you subtract $b$ and divide by $a$, which is $x = \frac{c - b}{a}$. It's just undoing things in reverse order, like taking off your shoes before your socks. This is the move every later lesson builds on.

Why "same thing to both sides" works. An equation says two quantities are equal. If you add 7 to both, or divide both by 3, they're still equal, so the new equation has exactly the same solutions as the old one. The only thing you can't do is multiply or divide by zero.

Strategy:

1. Clear fractions by multiplying both sides by the least common denominator.
2. Distribute and combine like terms on each side.
3. Move variable terms to one side and constants to the other.
4. Divide by the coefficient.
5. Check in the original equation.

Equations with variables in denominators (rational equations). Multiply through by the LCD, solve, and then reject any answer that makes an original denominator zero. Here's why that's needed: if the LCD contains $x$, you might have multiplied both sides by zero without noticing, and that can sneak in a fake solution.

Three outcomes. A linear equation has one solution, no solution, or every number as a solution. If the $x$'s cancel and you're left with something false like $0 = 5$, nothing works. If you're left with something always true like $0 = 0$, everything does.

Converting temperatures, splitting a bill, finding when two phone plans cost the same: everyday questions reduce to linear equations.