CDS Mathematics · Linear Equations

Solving Two Equations

Two linear equations in two unknowns meet in one point, in no point, or in every point, and the ratios of their coefficients tell you which.

All Linear Equations notes CDS Mathematics strategy

Why this matters

Ten PYQs, one of them HARD. Six ask for the solution — usually by adding and subtracting the equations, or by putting u = 1/x when each term is divided by xy. Four ask only whether a solution exists, which the coefficient ratios settle without solving.

Concept 1 of 2: Elimination and substitution

Add or subtract the equations so one unknown cancels. When the coefficients are swapped (

65x - 33y

and

33x - 65y

), adding and subtracting give

x + y

and

x - y

at once.

Definition

Eliminate: scale one equation so a variable's coefficients match, then subtract.
Swapped coefficients: add the equations and subtract them.
Terms like $7xy$ on one side: divide by $xy$ and put $u = \dfrac1x$ , $v = \dfrac1y$ .
A combination asked for ( $2n + 4e$ ) can come from a multiple of each equation without finding every unknown.

Swapped coefficients

ax + by = c,\; bx + ay = d \;\Rightarrow\; x + y = \dfrac{c + d}{a + b}

Worked example

Solve

47x + 31y = 125

and

31x + 47y = 109

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2018 · CDS (I) 2018 — Elementary Mathematics · Q7Moderate

Example 1 · Linear Equations · Solving Linear Systems

65x - 33y = 97

and

33x - 65y = 1

, then what is

xy

equal to?

Dividing by xy drops x = y = 0

Putting

u = \tfrac1x

assumes

x, y \ne 0

x = y = 0

also satisfies equations like

3(2u + v) = 7uv

; the options usually exclude it, but check.

Drill 5 more on elimination and substitution

Concept 2 of 2: One solution, none, or infinitely many

Two lines either cross, run parallel, or lie on top of each other. Parallel means the

x

and

y

coefficients are in the same ratio; on top means the constants are in that ratio too.

Definition

For $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$ :

$\dfrac{a_1}{a_2} \ne \dfrac{b_1}{b_2}$ : exactly one solution.
$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \ne \dfrac{c_1}{c_2}$ : no solution (inconsistent).
$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$ : infinitely many.
Equivalently, one solution exactly when $a_1b_2 - a_2b_1 \ne 0$ .

Unique solution

a_1b_2 - a_2b_1 \ne 0

Worked example

For what

k

3x + ky = 5

and

6x + 8y = 11

have no solution?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2017 · CDS (I) 2017 — Elementary Mathematics · Q47Easy

Example 2 · Linear Equations · Solving Linear Systems

The system of equations

2x + 4y = 6

and

4x + 8y = 8

Both signs of k

49 - k^2 \ne 0

rules out

k = 7

AND

k = -7

. An option saying '

k \ne 7

' is a correct consequence but not the whole condition.

Drill 3 more on one solution, none, or infinitely many

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)

Elimination and substitution
Swapped coefficients
$ax + by = c,\; bx + ay = d \;\Rightarrow\; x + y = \dfrac{c + d}{a + b}$
One solution, none, or infinitely many
Unique solution
$a_1b_2 - a_2b_1 \ne 0$

Watch out for (2)

Dividing by xy drops x = y = 0
→ Elimination and substitution
Both signs of k
→ One solution, none, or infinitely many

Test yourself on Linear Equations

15 past CDS questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Linear Equations chapter test Drill all 10 questions