Elementary algebra — simple equations

Every formula you touch as a power engineer — heat balances, pump head, steam tables, code wall-thickness — is an equation with one unknown you must isolate and solve. Get the algebra wrong and the boiler relief setting, the makeup-water rate, or the safe shell thickness comes out wrong. The chief's rule: rearrange the formula before you plug in numbers. It makes errors easier to catch and keeps your units honest.

What an equation is. An equation is a statement of equality: a quantity (or group of quantities) on the left equals a quantity on the right, joined by =. A simple equation has only one unknown. The cardinal rule is this: whatever arithmetic operation you perform on one side, you must perform identically on the other side, or the two sides are no longer equal. Add 2 to the left, add 2 to the right.

Transposing terms (addition and subtraction). Terms are quantities separated by + or - signs. To move a term to the other side of the equation, transfer it and change its sign: + becomes -, and - becomes +.

Worked example — transposing a term. Solve $0.004 + D = 0.0042$. Transpose the $0.004$ (it becomes $-0.004$): $D = 0.0042 - 0.004 = 0.0002$.

Transposing factors (multiplication and division). When the unknown is tied up in multiplication or division, move the factor across the division line. A factor multiplying on one side moves to divide on the other side, and vice versa. This is the basis of cross-multiplication: in $\frac{R_1}{R_2} = \frac{R_3}{R_4}$, the bottom of each side moves diagonally to the top of the other side, giving $R_1 R_4 = R_2 R_3$.

Worked example — cross-multiply (combined-gas form). Given $\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$, find $T_2$ when $P_1 = 100$, $V_1 = 2$, $T_1 = 300$, $P_2 = 150$, $V_2 = 3$. Cross-multiply to isolate $T_2$:

$T_2 = \frac{T_1 P_2 V_2}{P_1 V_1}$

Substitute: $T_2 = \frac{300 \times 150 \times 3}{100 \times 2} = \frac{135000}{200} = 675$.

When a term contains the unknown as a factor. If the unknown is a factor within a term, isolate that whole term first (using the rule for terms), then isolate the factor. Do not split the factor out while the term is still buried.

Worked example — term with a factor. Solve $7.5 = 36 - 4.5p$. You cannot move $4.5$ directly because $4.5p$ is a single term being subtracted. Transpose $36$ first, then divide: $4.5p = 36 - 7.5 = 28.5$, so $p = \frac{28.5}{4.5} = 6.33$.

Equations with roots and powers. To undo a square root, square both sides; to undo a square (the unknown is squared), take the square root of both sides. Whatever you do to one side, do to the other.

Worked example — roots and powers. Solve $A^2 = 121$. Take the root of both sides: $A = \sqrt{121} = 11$.

Worked example — chaining two techniques (factor, then root). Real formulas often need two moves in one problem. In $A = \pi r^2$, solve for $r$ when $A = 50.27$. First transpose the factor $\pi$ to isolate the squared term:

$r^2 = \frac{A}{\pi} = \frac{50.27}{3.1416} = 16.0$

Then take the square root of both sides to undo the power: $r = \sqrt{16.0} = 4.0$. Notice the order — clear the factor first so the power stands alone, then undo the power. Reverse the order and you cannot isolate $r$.

Fractional equations. Bring every fraction to the least common denominator (LCD), then drop the common denominator, leaving whole numbers.

Worked example — fractions. Solve $\frac{x}{3} + \frac{x}{6} = 5$. The LCD is 6: $\frac{2x}{6} + \frac{x}{6} = \frac{30}{6}$, so $2x + x = 30$, giving $3x = 30$ and $x = 10$.

Common misconceptions and exam traps. (1) Forgetting to change the sign when transposing a term — moving $+8$ across makes it $-8$, every time. (2) Half-transposing a term like $4.5p$ — isolate the whole term before splitting out the factor. (3) Doing the operation on only one side — the equality breaks instantly. (4) Substituting before rearranging — transpose symbolically first; it is cleaner and less error-prone, especially in multi-step thermodynamics and code calculations. (5) In a two-technique problem (factor then root), undoing the power before clearing the factor — clear the factor first.

Source: PanGlobal Third Class, Part A1 (Applied Mathematics); SOPEEC 3rd Class Paper 3A1.