Translating situations into algebraic equations

Algebra • Solving equations and inequalities

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Interpret solutions that contradict stated constraints, for example a negative age.

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Reject solutions that violate constraints and state that no valid solution exists under given conditions if necessary.

Key concepts

What you'll likely be quizzed about

Variables and expressions

An unknown quantity receives a single-letter symbol called a variable. Expressions combine variables, numbers and operations to represent procedures, for example 'three more than x' becomes x + 3. Limiting factors include domain restrictions; state whether a variable must be integer, non-negative, or within a specified range.

Translating phrases into algebra

Key words map to operations: 'sum' or 'total' indicates addition, 'difference' indicates subtraction, 'product' indicates multiplication, and 'quotient' indicates division. Phrases that describe relationships become equations by inserting an equals sign where a condition holds, for example 'twice a number is five more than the number' becomes 2x = x + 5. Clear parsing of the sentence prevents common translation errors.

Forming single equations

A condition or procedure that links quantities yields an equation. Identify unknowns, write expressions for each part, and set expressions equal when the problem states an equality or condition. Include given constraints explicitly, for example 'number is positive' or 'age is an integer', before solving.

Deriving and solving simultaneous equations

Two related conditions about two unknowns create a pair of simultaneous equations. Use substitution or elimination to obtain values for both variables. Verify each solution in both original equations to confirm consistency and reject extraneous roots.

Solving procedures and checking answers

Algebraic solution follows inverse operations and maintenance of equality. After isolation of the variable, simplify and solve, then substitute back into the original expression or both equations. Interpretation considers units, domain restrictions and the realistic feasibility of the numeric result.

Modelling limitations and assumptions

Translate conditional words like 'exactly', 'at least', 'more than', and 'less than' into appropriate equality or inequality signs. Explicitly state assumptions used when forming equations, such as continuous versus integer values, and adjust the final interpretation if assumptions restrict acceptable solutions.

Key notes

Important points to keep in mind

Assign a variable clearly and state any domain restrictions (integer, positive).

Translate phrase parts separately, then combine into one expression or equation.

Map key words to operations: sum=+, difference=-, product=×, quotient=÷.

Form an equation only when the problem gives an equality or condition to satisfy.

Choose substitution when one equation isolates a variable easily; choose elimination when coefficients match or scale easily.

Always substitute solutions back into original equation(s) to check correctness.

State and enforce constraints from context, reject solutions that contradict them.

Use consecutive-number models like n, n+1 for ordered integer situations.

Label units when quantities involve measurements to aid interpretation.

Translate inequalities carefully: 'at least' → ≥, 'at most' → ≤, 'more than' → >, 'less than' → <.