Linear Equations Notes — Sec 2 Math

Rules of balance (properties of equality)

$a = b \;\Rightarrow\; a + c = b + c,\quad a - c = b - c,\quad ac = bc,\quad \frac{a}{c} = \frac{b}{c}\ (c \neq 0)$

Whatever you do to one side, you must do to the other. Dividing by zero is undefined, and multiplying both sides by zero turns everything into 0 = 0.

Final solving step

$ax = b \;\Rightarrow\; x = \frac{b}{a}$

a is the coefficient of x, a fixed non-zero number, so dividing both sides by it is always allowed. Check the answer by substituting it into the original equation.

Cross-multiplication

$\frac{4}{x+1} = \frac{2}{x-3} \;\Rightarrow\; 4(x-3) = 2(x+1)$

Use when there is a fraction on each side: multiply both sides by both denominators at once. State the restrictions (here x ≠ −1 and x ≠ 3) before solving.

Expanding with a negative multiplier

−2(x + 3) = −2x − 6

The −2 multiplies every term in the bracket, sign included. Write the expansion line in full.

Consecutive integers

n, n + 1, n + 2 (consecutive integers); n, n + 2, n + 4 (consecutive even or odd integers)

Let n be the smallest integer. For even or odd integers the gap must be constant and equal to 2. Whether they are even or odd comes from the value of n you find.

Age problems moving forward in time

(f + 5) = 3(s + 5)

f and s are the father's and son's ages now. In 5 years' time every person is 5 years older, so add 5 to both ages, not just one.