Expressions, equations and systems

Foundations: letters, operations and equality

A letter may represent an unknown value or a varying value. In the expression 4x + 3, the notation 4x means 4 × x. If x = −2, the expression's value is 4(−2) + 3 = −5. An expression alone is neither true nor false. By contrast, 4x + 3 = 15 is an equation: it asks for which x values the two sides are equal.

Like terms have the same variable part. Therefore, 5x + 2x = 7x, but 5x + 2x² does not reduce to 7x³. The distributive property allows 3(x − 4) = 3x − 12. A minus sign before parentheses means multiplying every term by −1: −(2x − 7) = −2x + 7. This small review prevents many later errors.

An identity holds for every permitted value: 2(x + 3) = 2x + 6. A conditional equation holds only on part of the domain: 2x + 6 = 10 requires x = 2. Simplifying an equality may produce a truth such as 0 = 0, indicating all domain values, or a contradiction such as 0 = 5, indicating no solutions.

From sum to product: factoring

Factoring means writing an expression as a product. It is the distributive property read backwards: 6x² − 9x = 3x(2x − 3). The common factor includes the largest convenient numerical factor and the smallest power of each letter present in every term. To check, multiply again: 3x × 2x = 6x² and 3x × (−3) = −9x.

Special products also work both ways. We have (a + b)² = a² + 2ab + b² and (a − b)(a + b) = a² − b². The middle term of the square comes from the two products ab and ba; it cannot disappear. For x² − 7x + 12, we seek two numbers whose sum is −7 and product is 12: they are −3 and −4, so the expression is (x − 3)(x − 4).

When a product is zero, at least one factor is zero. Thus, (x − 3)(x − 4) = 0 has solutions 3 and 4. This rule depends on the product equalling zero. From (x − 3)(x − 4) = 6, we cannot conclude that one factor is 6; the equation must be rearranged.

Solve while preserving the original condition

Adding or subtracting the same number on both sides preserves equality. Multiplying or dividing both by a nonzero number does too. The informal expression “move it to the other side” hides these operations: in 7x − 5 = 23, we add 5 to both sides, obtaining 7x = 28, and divide both by 7 to reach x = 4.

If denominators are present, first record forbidden values. In (x + 2)/(x − 1) = 4, we need x ≠ 1. Multiplying by x − 1 within that domain gives x + 2 = 4x − 4; thus x = 2, which is permitted. Cancelling a factor does not restore a point where the original expression did not exist.

Nor should we divide by an expression that may be zero without studying that case. In x(x − 5) = 0, dividing by x would lose the solution x = 0. Factoring preserves both possibilities: x = 0 or x = 5. Finally, substituting into the original equation checks each candidate and helps detect sign or domain errors.

Inequalities: why the sign reverses

An inequality compares values through <, >, ≤ or ≥. Adding the same number to both sides preserves order: if a < b, then a + c < b + c. Multiplying by a positive number also preserves order. Multiplying by a negative number reverses it. To see why, start with 2 < 5: the opposites are −2 and −5, and −2 > −5, because the number line has been reflected about zero.

In general, if a < b, the difference b − a is positive. For c < 0, the product c(b − a) is negative, that is, cb − ca < 0; therefore cb < ca. The reversal has a mathematical reason and is not a ritual reserved for a worksheet.

For example, −3x + 2 ≤ 11 implies −3x ≤ 9 and, dividing by −3, x ≥ −3. The point −3 is included because the inequality allows equality. The solution is a set of numbers, not just one number: [−3, +∞[. If you multiply an inequality by an expression of unknown sign, you need to split into cases; the sign cannot be guessed.

Square roots and extraneous candidates

A quadratic equation has the form ax² + bx + c = 0, with a ≠ 0. Some can be solved by factoring and others by isolating a square. If (x − 2)² = 25, then x − 2 = 5 or x − 2 = −5, so x = 7 or x = −3. The principal root √25 is 5; the two signs of the unknown produce the two solutions.

Squaring both sides requires care. From a = b follows a² = b², but a² = b² also allows a = −b. The transformation may therefore produce extraneous candidates. In √(x + 6) = x, the domain and nonnegativity of the first side require x ≥ 0. Squaring gives x + 6 = x², or (x − 3)(x + 2) = 0.

The candidates are 3 and −2, but only 3 satisfies the original equation: √9 = 3. For −2, √4 = 2, different from −2. The calculation did not “manufacture” a valid solution; it produced a more permissive condition requiring verification. This distinction reappears with logarithms and other nonreversible operations.

Two conditions on the same unknowns

A system requires every equation to hold at the same time. The pair (x, y) is ordered: swapping coordinates may change the answer. In substitution, we isolate one letter in one equation and put the resulting expression into the other. In elimination, we add equations after suitable multiplication to cancel one letter. Both techniques seek the same set of pairs.

If x + y = 9 and 2x − y = 6, adding the equalities gives 3x = 15. We find x = 5 and, in the first equation, y = 4. Elimination worked because y and −y sum to zero. Verification must use both equations: a pair satisfying only one condition does not solve the system.

In the plane, each linear equation whose two coefficients are not simultaneously zero represents a line. Intersecting lines give one solution; distinct parallel lines give none; coincident lines give infinitely many. When modelling, define letters and units before equations. If they represent numbers of objects, negative or noninteger solutions may be incompatible with the context even if they satisfy the calculations.

Worked examples, without skipping steps

Follow the reasoning first. Then cover the solution and try to reconstruct it in your own words.

Now try it yourself

There is no timer. Write what you know, choose a relationship and justify each step. Hints and solutions are available whenever you need them.

Mistakes worth avoiding

Moving a term to the other side is a rule without justification.

The move summarises adding or subtracting the same term on both sides.

Dividing an inequality by −2 preserves the sign.

The sign reverses because multiplying by a negative number reverses order on the number line.

Every result after squaring is a solution.

They are candidates; they must be checked in the original equation and respect the domain.

A system is solved once x is found.

The complete pair must be found and all equations checked.