Forming & Solving Equations

Step 1: Translation

• Identify the unknown quantity and assign it a letter, such as $x$. Translate keywords like 'sum' ($+$), 'difference' ($-$), 'product' ($imes$), and 'is' ($=$) into algebraic symbols.

Step 2: Construction

• Build the equation by relating the algebraic terms. For example, if a total length is composed of three equal segments and a remainder, the equation might look like $3x + r = L$.

Step 3: Isolation

• Use the reverse order of operations (SAMDEB) to isolate the variable. Start by moving terms added or subtracted from the variable's side, then address coefficients through division or multiplication.

Step 4: Verification

• Always substitute the final value back into the original context of the problem. This ensures that the solution makes sense both algebraically and logically within the scenario provided.

It is vital to distinguish between an expression and an equation. An expression like $3x + 5$ cannot be solved because it has no defined value; an equation like $3x + 5 = 20$ provides the necessary constraint to find $x$.

Distinguish between linear and non-linear equations. Linear equations involve variables raised only to the power of 1 (e.g., $ax + b = c$), resulting in a single solution, whereas non-linear equations (like quadratics) may have multiple solutions.

• Read the Final Question: Exams often ask for something other than the variable itself, such as 'the area of the shape' or 'the age of the older person.' After finding $x$, ensure you perform the final calculation required.

• Check for Units: If a problem involves physical quantities like meters or dollars, ensure your final answer includes the correct units and that all initial values were converted to a consistent system before forming the equation.

• Show Intermediate Steps: Examiners award marks for the process of isolation. Even if the final numerical answer is wrong, showing the correct inverse operations can secure partial credit.

• Sanity Checks: If you are solving for a person's age and get a negative number or a fraction like $45.2$, re-examine your equation setup, as these results are usually unrealistic for the context.

• The 'Less Than' Trap: Students often translate '5 less than $x$' as $5 - x$ instead of the correct $x - 5$. The order of subtraction is critical and is a frequent source of error in word problems.

• Incomplete Operations: When multiplying or dividing an entire side of an equation, students often forget to apply the operation to every term. For example, in $\frac{x + 4}{2} = 10$, the entire numerator is affected by the division.

• Sign Errors during Transfer: When moving a term to the other side of the equals sign, its sign must change (e.g., $+5$ becomes $-5$). Forgetting this change disrupts the balance of the equation.