Linear Equations

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In 30 seconds

What: Linear equations in one and two variables, simultaneous systems, consistency criteria, graphical and algebraic methods, and word problems (age, digit, mixture, two-pipe rate).
Why it matters: CDS papers from 2000 to 2023 average 3–5 questions per sitting — and the consistency / unique-solution criterion is a high-frequency MCQ trap.
Key fact: Two linear equations $a_1 x + b_1 y + c_1 = 0$ and $a_2 x + b_2 y + c_2 = 0$ have a unique solution iff $\tfrac{a_1}{a_2} \neq \tfrac{b_1}{b_2}$; no solution iff $\tfrac{a_1}{a_2} = \tfrac{b_1}{b_2} \neq \tfrac{c_1}{c_2}$; infinitely many iff all three ratios are equal.

Linear Equations is the algebra chapter that translates word problems into manageable two-line systems. CDS tests both the algebra (substitution, elimination, cross-multiplication) and the geometry (graph of $ax + by + c = 0$ is a line). The high-yield reflexes: spot the consistency criterion, set up word problems with clean variable choices, and solve in 30 seconds.

This page is built from CDS Previous Year Questions across 2000–2023, plus NCERT Class 9 Linear Polynomials and Class 10 Pair of Linear Equations. Pair with Basic Operations and Factorisation and Quadratic Equations.

What This Topic Covers

CDS scope: (1) one-variable — $ax + b = 0$ and standard rearrangement; (2) two-variable systems — substitution, elimination, cross-multiplication; (3) consistency — unique, no, or infinite solutions; (4) graphical interpretation — intersecting, parallel, coincident lines; (5) word problems — age, digit, mixture, pipe, two-variable distance/time; and (6) determinant / Cramer's rule for 2×2 systems.

Why This Topic Matters

3–5 CDS questions per paper, mix of pure algebra and word problems.
The consistency criterion gives instant elimination of distractors.
Linear-equation word problems are formula-light — they reward clean variable setup.

Exam Pattern & Weightage

Year / Paper	No.	Subtopics Tested
2007-II	3	Two-variable, consistency, age problem
2009-II	3	Substitution, digit problem
2010-I/II	4	Consistency criterion, elimination
2011-I/II	3	Cross-multiplication, word
2013-I/II	4	Two-variable systems, age
2014-I/II	3	Word problems, consistency
2015-I/II	3	Substitution, digit reversal
2017-I/II	4	Unique solution, age and digit
2019-II	3	Cross-multiplication, mixture
2020-I/II	4	Word problems, consistency
2021-I/II	3	Mixed
2022-I / 2023-I	3	Mixed application

⚡ CDS Alert

For two equations to have no solution (parallel lines), the ratios of $a, b$ must be equal but the ratio of $c$ must differ. This single test eliminates 2-3 options instantly in CDS consistency questions.

Core Concepts

One-Variable Linear Equation

Of the form $ax + b = 0$. Solution: $x = -b/a$ (provided $a \neq 0$). Trivial; CDS rarely asks this directly.

Two-Variable Linear System

Standard Form $$a_1 x + b_1 y + c_1 = 0 \qquad a_2 x + b_2 y + c_2 = 0$$

Consistency Criteria Unique solution: $\tfrac{a_1}{a_2} \neq \tfrac{b_1}{b_2}$ (intersecting lines).
No solution: $\tfrac{a_1}{a_2} = \tfrac{b_1}{b_2} \neq \tfrac{c_1}{c_2}$ (parallel lines).
Infinite solutions: $\tfrac{a_1}{a_2} = \tfrac{b_1}{b_2} = \tfrac{c_1}{c_2}$ (coincident lines).

Three Solving Methods

Substitution: solve one equation for one variable; substitute into the other.
Elimination: multiply equations by appropriate factors to make one variable's coefficients equal; subtract.
Cross-Multiplication: for $a_1 x + b_1 y + c_1 = 0$ and $a_2 x + b_2 y + c_2 = 0$:

Cross-Multiplication Formula $$\frac{x}{b_1 c_2 - b_2 c_1} = \frac{y}{c_1 a_2 - c_2 a_1} = \frac{1}{a_1 b_2 - a_2 b_1}$$

Word Problem Setups

Age: "Father is twice as old as son. In 5 years, the ratio will be 5:3." Let father = $f$, son = $s$. Equations: $f = 2s$; $(f + 5)/(s + 5) = 5/3$.

Digit: "A two-digit number has tens digit $t$, units digit $u$. Number = $10t + u$; reverse = $10u + t$; difference $= 9(t - u)$ — always a multiple of 9."

Worked Examples

Example 1 — Substitution (2007-II)

Q: Solve $2x + 3y = 12$ and $x - y = 1$.

From second: $x = y + 1$. Substitute: $2(y + 1) + 3y = 12 \implies 5y = 10 \implies y = 2$.
Then $x = 3$. Solution: $(3, 2)$.

Example 2 — Consistency (2010-II)

Q: For what $k$ does the system $2x + 3y = 5$, $4x + 6y = k$ have infinite solutions?

For infinite solutions: $\tfrac{2}{4} = \tfrac{3}{6} = \tfrac{5}{k}$.
Two ratios are $\tfrac{1}{2}$; set $\tfrac{5}{k} = \tfrac{1}{2} \implies k = 10$.

Example 3 — Age Problem (2013-I)

Q: A father is 3 times as old as his son. After 12 years, the father will be twice as old. Find present ages.

Let son = $s$. Father = $3s$. After 12 years: father = $3s + 12$, son = $s + 12$.
$3s + 12 = 2(s + 12) \implies 3s + 12 = 2s + 24 \implies s = 12$.
Father = 36, son = 12.

Example 4 — Digit Reversal (2015-I)

Q: A two-digit number's digits sum to 9. If the digits are reversed, the new number is 27 more. Find the original.

Let digits be $t$ (tens) and $u$ (units). Original = $10t + u$, reverse = $10u + t$.
$t + u = 9$; $(10u + t) - (10t + u) = 27 \implies 9u - 9t = 27 \implies u - t = 3$.
Solve: $t = 3, u = 6$. Original = 36.

Example 5 — Elimination (2017-II)

Q: Solve $3x + 4y = 11$ and $2x - y = 4$.

From second: $y = 2x - 4$. Substitute: $3x + 4(2x - 4) = 11 \implies 11x - 16 = 11 \implies x = 27/11$.
Then $y = 2(27/11) - 4 = 54/11 - 44/11 = 10/11$. Solution: $(27/11, 10/11)$.

Example 6 — No Solution (2019-II)

Q: For what $k$ does the system $kx + 2y = 5$, $3x + y = 4$ have no solution?

No solution: $\tfrac{k}{3} = \tfrac{2}{1} \neq \tfrac{5}{4}$.
From first ratio: $k = 6$. Check inequality: $\tfrac{5}{4} \neq 2$, holds. ✓

Example 7 — Distance/Time System (2020-I)

Q: A boat travels 30 km downstream in 3 hours and 18 km upstream in 3 hours. Find still-water and stream speeds.

Let still-water speed = $u$, stream = $s$. Downstream = $u + s = 10$. Upstream = $u - s = 6$.
Adding: $2u = 16 \implies u = 8$. Subtracting: $2s = 4 \implies s = 2$.

How CDS Tests This Topic

Five archetypes: (1) solve a system algebraically (substitution or elimination), (2) consistency criterion — find $k$ for unique/no/infinite solutions, (3) age word problem, (4) digit-reversal problem, (5) two-variable distance/time/work system.

Exam Shortcuts (Pro-Tips)

Shortcut 1 — Consistency Reflex

Memorise: unique $\Leftrightarrow \tfrac{a_1}{a_2} \neq \tfrac{b_1}{b_2}$; no solution $\Leftrightarrow$ equal but $c$ differs; infinite $\Leftrightarrow$ all three equal. This single line solves the consistency archetype.

Shortcut 2 — Digit-Number Identity

Two-digit $\overline{tu} = 10t + u$. Reverse: $10u + t$. Difference: $9(t - u)$ — always a multiple of 9.

Shortcut 3 — Elimination by Symmetry

For $u + s = a$ and $u - s = b$ (boat/stream, train relative speeds): add for $2u = a + b$, subtract for $2s = a - b$. One step each.

Shortcut 4 — Cramer's Rule for 2×2

For $a_1 x + b_1 y = c_1$ and $a_2 x + b_2 y = c_2$: $D = a_1 b_2 - a_2 b_1$; $x = (c_1 b_2 - c_2 b_1)/D$; $y = (a_1 c_2 - a_2 c_1)/D$. Memorise the determinant pattern.

Shortcut 5 — Age "Constant Difference" Trick

For age problems, the difference between two ages is constant. So if father is currently 30 years older than son, that gap stays at 30 forever.

Common Question Patterns

Pattern 1 — Solve a System

Substitution or elimination. Pick whichever has a clean variable to isolate.

Pattern 2 — Consistency Criterion

Apply the three-ratio test. Solve for the parameter $k$ that makes the system unique / impossible / coincident.

Pattern 3 — Age Problem

Set present ages as variables. Translate "after $n$ years" or "before $n$ years" into linear relations. Solve.

Pattern 4 — Digit Reversal

Define digits as variables. Use sum-of-digits and reverse-minus-original conditions. The 9-multiple identity often simplifies.

Pattern 5 — Two-Variable Word Setup

Mixture, distance, work problems where two unknowns satisfy two conditions. Translate, solve.

Preparation Strategy

Week 1. Master substitution and elimination on 20 systems. Memorise the consistency criterion. Practice 15 age and digit problems with clean variable setups.

Week 2. Two-variable distance/work/mixture problems. Cramer's rule for occasional speed wins. Layer with Basic Operations and Quadratic Equations.

Mock testing. Use CDS mock tests. Most slip-ups: dropping a negative sign or forgetting to multiply through. Drill until clean.

Drill Linear Equations

CDS mocks with consistency criteria, age, and digit problems. Five archetypes — three solving methods — reflex.

Start Free Mock Test

Frequently Asked Questions

When does a 2x2 linear system have a unique solution?

When the coefficient ratios differ: $\tfrac{a_1}{a_2} \neq \tfrac{b_1}{b_2}$. Geometrically, the two lines are not parallel — they intersect at one point.

When does a system have no solution?

When the coefficient ratios are equal but the constants ratio differs: $\tfrac{a_1}{a_2} = \tfrac{b_1}{b_2} \neq \tfrac{c_1}{c_2}$. The lines are parallel but distinct.

When does a system have infinitely many solutions?

When all three ratios are equal: $\tfrac{a_1}{a_2} = \tfrac{b_1}{b_2} = \tfrac{c_1}{c_2}$. The two equations represent the same line.

What's the fastest method for solving a 2x2 system?

If one variable has coefficient 1 or −1 in one equation, use substitution. Otherwise, elimination by multiplying through. Cramer's rule is fastest for systems with clean numerical coefficients.

How do I set up an age problem?

Let present ages be variables. Translate each time-clause: "Now", "after $n$ years" (add $n$ to each), "before $n$ years" (subtract $n$). Each clause gives one equation. Two unknowns ⇒ two equations needed.

What's the digit-reversal identity?

For a two-digit number with tens digit $t$ and units digit $u$: number = $10t + u$; reverse = $10u + t$; their difference $= 9(t - u)$. Always a multiple of 9 — a handy elimination tool in CDS MCQ options.

Which CDS Maths topics connect to Linear Equations?

Basic Operations — algebraic manipulation. Quadratic Equations — sometimes a linear-quadratic system appears. TSD and Time and Work — two-variable systems appear in their word problems.