Class 10 Maths Chapter 3 Linear Equations Formula Sheet

The Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables formula sheet puts every key result on one page. It covers the general form of a pair, the meaning of a solution, the ratio (consistency) test for the number of solutions, and the graphical, substitution and elimination methods. Built for the 2026-27 CBSE syllabus, it is made for a quick night-before revision.

• All core formulas in one place, each with a plain-English meaning.
• Ratio test for unique, infinite or no solution straight from the coefficients, plus when each method is fastest.
• Method toolkit: graphical, substitution and elimination steps, with one worked example each.

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Complete Formula List

The table below lists every named result you need. The chapter rests on two ideas: a pair of lines can meet, overlap or stay apart, giving a unique, infinite or no solution; and the ratio test on the coefficients tells you which case you are in, while three methods find the solution.

General Form and Solution of a Pair

A pair of linear equations in two variables is two equations of the form a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0. Each is degree 1, so each draws a straight line. A solution is a pair (x, y) that makes both true, that is the point of intersection.

• Consistent: at least one solution, either unique or infinitely many.
• Inconsistent: no solution (parallel lines).
• Dependent: the two equations are the same line, so infinitely many solutions; always consistent.

First write each equation in "= 0" form and read off a, b and c with their signs; a sign slip breaks the ratio test below.

Ratio Test for the Number of Solutions

This is the most-tested idea in the chapter. Comparing the coefficient ratios predicts the number of solutions without a graph, in three cases. The c-ratio separates parallel from coincident.

For x + 2y − 4 = 0 and 2x + 4y − 12 = 0: a₁/a₂ = 1/2, b₁/b₂ = 1/2, c₁/c₂ = 1/3. The first two match but the c-ratio differs, so the lines are parallel with no solution. Reduce each fraction first.

Graphical Method: Three Cases

Draw both lines on the same axes; where, or whether, they meet decides the number of solutions. To plot a line, find two solutions and join them.

• Intersecting lines: cross at one point, so a unique solution (consistent).
• Coincident lines: lie on top of each other, so infinitely many solutions (dependent).
• Parallel lines: never meet, so no solution (inconsistent).

In a graph question, one crossing point means one solution, overlap means infinitely many, and a clear gap means none.

Substitution Method: Formula & Steps

Substitution makes one variable the subject, then puts that expression into the other equation. It works for every consistent pair and is easiest when a coefficient is 1 or −1.

• Step 1: from one equation, write one variable in terms of the other, e.g. x = (c − by) / a.
• Step 2: substitute into the other equation to get one equation in one variable, and solve.
• Step 3: put that value back into the Step-1 expression for the second variable.

For x + 2y = 3 and 7x − 15y = 2, the first gives x = 3 − 2y. Substituting, 21 − 29y = 2, so y = 19/29 and x = 49/29. A true statement with no variable left means infinitely many solutions; a false one means no solution.

Elimination Method: Formula & Steps

Elimination scales the equations so one variable's coefficients match in size, then adds or subtracts to cancel it. It is often quicker than substitution when no coefficient is 1.

For 9x − 4y = 2000 and 7x − 3y = 2000, multiply by 3 and 4 to match y, then subtract to get x = 2000 and y = 4000, so the solution is (2000, 4000). Scale the constant c by the same factor as the rest.

Equations Reducible to Linear Form

Some pairs become linear after a smart substitution: replace a repeated awkward block with a new letter, solve the linear pair, then undo the substitution.

• Reciprocal pairs: for equations in 1/x and 1/y, put u = 1/x and v = 1/y to make the pair linear in u and v.
• Solve and recover: find u and v, then recover x = 1/u and y = 1/v.

The mark students lose most often is forgetting to undo the substitution. After finding u and v, convert back to x and y.

How to Use This Formula Sheet

• Night-before revision: state each rule, especially the three ratio conditions, without your notes.
• During practice: keep the PDF open to look up the ratio test and method steps.
• For graph questions: one crossing point means a unique solution, overlap means infinitely many, a gap means none.

Weightage in the CBSE Board Exam

This chapter sits in the Algebra unit. The table shows where its topics fit among the high-frequency question types.

Across recent CBSE papers, this chapter usually carries about 5 to 6 marks, often through a word problem.

Common Mistakes With These Formulas

More Resources for This Chapter

Use this formula sheet with the resources below.

NCERT Formula Sheets for Class 10 Maths: All Chapters