Is it correct to say: Comparing ratios when one equation is ax + by + c = 0 and the other is ax + by = c?

Bring both to the same form first. Otherwise c₁/c₂ gets the wrong sign and the conclusion can flip.

Is it correct to say: Concluding 'no solution' from a₁/a₂ = b₁/b₂ alone?

You must also check c₁/c₂. If it is equal too, the lines coincide and there are infinitely many solutions.

Is it correct to say: Dropping the negative sign of a coefficient when writing b₁/b₂?

For 2x − 3y = 5, b = −3, not 3. Write each coefficient with its sign before forming ratios.

Is it correct to say: Multiplying only some terms of an equation during elimination?

Multiply EVERY term, including the constant on the right-hand side.

Is it correct to say: Keeping only one value of k when k² gives two?

Check each value against the remaining condition (≠ or =) and keep every value that works.

Is it correct to say: Stopping at x and y in a word problem?

Answer the question in words with units, e.g. 'One pen costs ₹15', and verify with both conditions.

📄 Formula sheet (PDF)

Pair of Linear Equations in Two Variables

Graphical meaning of a pair of linear equations, consistency conditions from coefficient ratios, and solution by substitution and elimination — NCERT Class 10 Maths Ch 3

📐 11 formulas✏️ 4 examples🎯 6 practice⚖️ Unit II (Algebra): 20 marks (with Ch 2, 4, 5)🏫 CBSE📚 Class 10✓ 2025–26 syllabus

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Board Exam Tips

→For 'consistent or inconsistent?' questions, write the three ratios a₁/a₂, b₁/b₂, c₁/c₂ explicitly, compare them, then name the lines (intersecting, coincident or parallel).
→Before comparing ratios, put BOTH equations in the same form (both ax + by + c = 0 or both ax + by = c).
→Use substitution when some variable has coefficient 1 or −1; otherwise use elimination.
→In word problems, begin with 'Let the cost of one pen be ₹x and of one pencil be ₹y' — defining variables with units earns marks.
→Always verify the solution by putting it back into BOTH original equations.
→For the graphical method, plot at least two points per line (a third as a check), label each line and read the solution from the point of intersection.

📐 Formulas(11)

General Form of a Pair

a_{1} x + b_{1} y + c_{1} = 0, a_{2} x + b_{2} y + c_{2} = 0

Symbol	Meaning
$x, y$	The two unknowns (variables)
$a_{1}, b_{1}, a_{2}, b_{2}$	Coefficients of x and y in the two equations
$c_{1}, c_{2}$	Constant terms

Axis Intercepts of a Line

a x + b y + c = 0 : x-intercept = - \frac{c}{a}, y-intercept = - \frac{c}{b}

Solution of the Pair

(x_{0}, y_{0}) is a solution ⟺ a_{1} x_{0} + b_{1} y_{0} + c_{1} = 0 and a_{2} x_{0} + b_{2} y_{0} + c_{2} = 0

Intersecting Lines — Unique Solution★ Board fav

\frac{a _{1}}{a _{2}} \neq = \frac{b _{1}}{b _{2}} \Rightarrow unique solution

Coincident Lines — Infinitely Many Solutions★ Board fav

\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} = \frac{c _{1}}{c _{2}} \Rightarrow infinitely many solutions

Parallel Lines — No Solution★ Board fav

\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} \neq = \frac{c _{1}}{c _{2}} \Rightarrow no solution

Substitution Method

y = - \frac{a _{1} x + c _{1}}{b _{1}} (b_{1} \neq = 0) \to substitute into the second equation

Elimination Method★ Board fav

a_{2} (a_{1} x + b_{1} y + c_{1}) - a_{1} (a_{2} x + b_{2} y + c_{2}) = 0 \Rightarrow (a_{2} b_{1} - a_{1} b_{2}) y = a_{1} c_{2} - a_{2} c_{1}

When Both Variables Disappear

0 = 0 \Rightarrow infinitely many solutions; 0 = k (k \neq = 0) \Rightarrow no solution

Two-digit Number Set-up

Number = 10 t + u, Reversed number = 10 u + t

Symbol	Meaning
$t$	Tens digit
$u$	Units digit

Age Problem Set-up

Age n years ago = x - n, Age n years hence = x + n

✏️ Solved Examples

1Solved Exampleeasy3 steps

Without solving, find whether the pair 6x − 9y + 4 = 0 and 4x − 6y + 7 = 0 is consistent or inconsistent.

Both equations are already in the form ax + by + c = 0. Compute the ratios.

\frac{a _{1}}{a _{2}} = \frac{6}{4} = \frac{3}{2}, \frac{b _{1}}{b _{2}} = \frac{- 9}{- 6} = \frac{3}{2}, \frac{c _{1}}{c _{2}} = \frac{4}{7}

2Solved Exampleboard5 steps

Five years ago, Asha was three times as old as her brother. Ten years from now, she will be twice as old as him. Find their present ages using the substitution method.

Let Asha's present age be x years and her brother's be y years. Five years ago:

x - 5 = 3 (y - 5) \Rightarrow x = 3 y - 10 (1)

3Solved Exampleboard4 steps

For what value(s) of k does the pair kx + 4y = 5 and 9x + ky = 8 have no solution?

For no solution we need a₁/a₂ = b₁/b₂ ≠ c₁/c₂. Both equations are in the form ax + by = c, so the ratios can be compared directly.

\frac{k}{9} = \frac{4}{k} \Rightarrow k^{2} = 36 \Rightarrow k = 6 or k = - 6

4Solved ExampleHOTS5 steps

Find a and b so that the pair 2x + 3y = 7 and (a + b)x + (2a − b)y = 21 has infinitely many solutions.

For infinitely many solutions all three ratios must be equal.

\frac{2}{a + b} = \frac{3}{2 a - b} = \frac{7}{21} = \frac{1}{3}

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Comparing ratios when one equation is ax + by + c = 0 and the other is ax + by = c
✓Bring both to the same form first. Otherwise c₁/c₂ gets the wrong sign and the conclusion can flip.
2
Concluding 'no solution' from a₁/a₂ = b₁/b₂ alone
✓You must also check c₁/c₂. If it is equal too, the lines coincide and there are infinitely many solutions.
3
Dropping the negative sign of a coefficient when writing b₁/b₂
✓For 2x − 3y = 5, b = −3, not 3. Write each coefficient with its sign before forming ratios.
4
Multiplying only some terms of an equation during elimination
✓Multiply EVERY term, including the constant on the right-hand side.
5
Keeping only one value of k when k² gives two
✓Check each value against the remaining condition (≠ or =) and keep every value that works.
6
Stopping at x and y in a word problem
✓Answer the question in words with units, e.g. 'One pen costs ₹15', and verify with both conditions.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
Solve by substitution: x + y = 14 and x − y = 4.
Q2
Solve by elimination: 3x + 4y = 10 and 2x − 2y = 2.
Q3
For what value of k does the pair 2x + 5y = 1 and 4x + ky = 3 have no solution?
Q4
Show that the pair x − 2y = 3 and 3x − 6y = 9 has infinitely many solutions.
Q5
The sum of the digits of a two-digit number is 11. The number formed by reversing the digits exceeds the original number by 27. Find the number.
Q6
2 adult tickets and 3 child tickets cost ₹520, while 1 adult ticket and 2 child tickets cost ₹300. Find the cost of each ticket.

📝 Notes

Pair of Linear Equations in Two Variables

Each linear equation in x and y is a straight line. Solving a pair means finding where the two lines meet, and there are only three possibilities.

Read the answer from the ratios

Compare a₁/a₂, b₁/b₂ and c₁/c₂:

Ratios	Lines	Solutions	Pair is
a₁/a₂ ≠ b₁/b₂	Intersecting	Exactly one	Consistent
a₁/a₂ = b₁/b₂ = c₁/c₂	Coincident	Infinitely many	Dependent (consistent)
a₁/a₂ = b₁/b₂ ≠ c₁/c₂	Parallel	None	Inconsistent

"Find k" questions are this table run backwards: choose the row you need, set the ratios equal (or unequal), solve for k, and check every value you get.

Choosing a method

Graphical: plot both lines and read off the point of intersection. Good for seeing the answer; less exact when the solution is not a whole number.
Substitution: best when one variable has coefficient 1 or −1, e.g. x − y = 4 gives x = y + 4 at once.
Elimination: best when no coefficient is 1. Multiply so that one variable has equal coefficients, then add or subtract.

If both variables disappear while solving, look at what is left: 0 = 0 means infinitely many solutions; 0 = 5 means none.