IMAT Mathematics · Equations and Inequalities

Linear Equations, Parameters and Systems

A linear equation reduces to ax = b, which has one solution, none or infinitely many; two linear equations in two unknowns are solved by substitution or elimination.

All Equations and Inequalities notes

Why this matters

A system of two equations appeared in 2015. The 2025 paper asked twice whether a linear equation with a parameter has one solution, none, or infinitely many.

Concept 1 of 2: Linear equations and equations with a parameter

Whatever you do to one side, do to the other, and a linear equation always ends as

ax = b

. Dividing by

a

is the last step, and it is allowed only when

a \ne 0

. When a letter (a parameter) sits in

a

, the case

a = 0

has to be checked separately.

Definition

Clear fractions by multiplying every term by the lowest common denominator, collect terms, and reach $ax = b$ . Then:

$a \ne 0$ : exactly one solution, $x = \frac{b}{a}$ . The equation is determined.
$a = 0$ and $b = 0$ : $0 = 0$ , every real $x$ is a solution. The equation is indeterminate (an identity).
$a = 0$ and $b \ne 0$ : $0 = b$ , no solution. The equation is impossible.

Solution of ax = b

ax = b,\; a \ne 0 \;\Rightarrow\; x = \frac{b}{a}

Worked example

(a) Solve

\dfrac{x - 1}{2} - \dfrac{x + 2}{3} = 1

. (b) Discuss

(k - 2)x = k^2 - 4

for every value of the parameter

k

Practice this conceptself-check · 4 quick reps

Zero coefficient: check the right-hand side

When the coefficient of

x

is zero, the equation is not automatically impossible. If the right-hand side is also zero, every

x

works. Only a zero coefficient with a non-zero right-hand side gives no solution.

Concept 2 of 2: Systems of two linear equations

Each linear equation in

x

and

y

is a straight line, and a solution of the system is a point on both lines. Two lines usually cross once; parallel lines never meet; and two equations that describe the same line share every point.

Definition

Substitution: make one unknown the subject of one equation and put it into the other.
Elimination: multiply the equations so one unknown has opposite coefficients, then add.
One solution when the coefficients of $x$ and $y$ are not in proportion.
No solution when they are in proportion but the constants are not (parallel lines): $x + 2y = 3$ , $2x + 4y = 7$ .
Infinitely many when the whole of one equation is a multiple of the other.
If a question asks only for $x + y$ or $x - y$ , adding or subtracting the equations may give it directly.

Two equations in two unknowns

\begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases}

Worked example

Solve

3x + 2y = 16

and

x - y = 2

Practice this conceptself-check · 4 quick reps

Proportional coefficients do not always mean no solution

If the

x

and

y

coefficients are in proportion, the lines are parallel. They have no common point only if the constants break the proportion; if the constants follow it too, the lines coincide and there are infinitely many solutions.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Linear equations and equations with a parameter
Solution of ax = b
$ax = b,\; a \ne 0 \;\Rightarrow\; x = \frac{b}{a}$
Systems of two linear equations
Two equations in two unknowns
$\begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases}$

Watch out for (2)

Zero coefficient: check the right-hand side
→ Linear equations and equations with a parameter
Proportional coefficients do not always mean no solution
→ Systems of two linear equations

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