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CDS Mathematics · Formula sheet

Linear Equations formulas

5 formulas and 5 common traps for CDS Mathematics Linear Equations, grouped by subtopic.

Full notes with worked examples

Solving Two Equations

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Elimination and substitution

Swapped coefficients

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

One solution, none, or infinitely many

Unique solution

a1b2−a2b1≠0a_1b_2 - a_2b_1 \ne 0

Common traps

Dividing by xy drops x = y = 0

Putting u=1xu = \tfrac1x assumes x,y≠0x, y \ne 0. x=y=0x = y = 0 also satisfies equations like 3(2u+v)=7uv3(2u + v) = 7uv; the options usually exclude it, but check.

Both signs of k

49−k2≠049 - k^2 \ne 0 rules out k=7k = 7 AND k=−7k = -7. An option saying 'k≠7k \ne 7' is a correct consequence but not the whole condition.

Word Problems

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From sentences to equations

Reversed digits

(10a+b)−(10b+a)=9(a−b)(10a + b) - (10b + a) = 9(a - b)

Common traps

The remainder must be smaller than the divisor

'Quotient 66, remainder 66' only works when the divisor exceeds 66. Check each remainder against its divisor before accepting a solution.

Age Problems

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Past, present and future ages

n years ago

R−n=k(S−n)R - n = k(S - n)

Common traps

Subtract from both ages

'Five years ago Ram was three times as old as Shyam' is R−5=3(S−5)R - 5 = 3(S - 5). Subtracting only from Ram's side gives the wrong equation.

Whole-Number Solutions

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Stepping through the solutions

All solutions

x=x0+bt,y=y0−atx = x_0 + bt, \quad y = y_0 - at

Common traps

Does zero count?

'Buy both apples and oranges' rules out zero of either; 'non-negative' allows it. The count changes by one or two at each end, which is exactly the spread of the options.

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