Pair of Linear Equations Class 10: All 5 Methods with Solved Examples

Pair of Linear Equations in Two Variables (NCERT Chapter 3) is one of the most important chapters in Class 10 Maths. It typically carries 6-8 marks in the CBSE board exam and appears across all question types — from MCQs to 5-mark word problems.

The chapter teaches you five different methods to solve a system of two equations with two unknowns. While you only need to master 2-3 methods for the exam, understanding all five gives you flexibility to choose the fastest approach for any given problem.

The general form of a pair of linear equations:

where $a_1, b_1, c_1, a_2, b_2, c_2$ are real numbers and $a_1^2 + b_1^2 \neq 0$, $a_2^2 + b_2^2 \neq 0$ (i.e., the coefficients of $x$ and $y$ are not both zero in either equation).

Let's go through each method with clear examples!

Before solving a system, it's crucial to know whether a solution even exists. This is determined by the consistency conditions, which are among the most frequently asked MCQ topics.

For the pair of equations $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$:

Case 1: Exactly one solution (Consistent — Intersecting lines)

Case 2: Infinitely many solutions (Dependent — Coincident lines)

Case 3: No solution (Inconsistent — Parallel lines)

The graphical method involves plotting both equations as straight lines on a coordinate plane and finding the point of intersection.

Steps:
1. Rewrite each equation in the form $y = mx + c$ (slope-intercept form).
2. Find at least two points for each line by substituting convenient values of $x$.
3. Plot the points and draw the lines.
4. The coordinates of the intersection point give the solution.

When to use: Primarily for understanding concepts and for 1-2 mark questions that ask you to identify the type of solution from a graph. For actual solving, algebraic methods are faster and more accurate.

The substitution method involves expressing one variable in terms of the other from one equation, and substituting it into the second equation.

Steps:
1. From one equation, express $y$ in terms of $x$ (or $x$ in terms of $y$).
2. Substitute this expression into the other equation.
3. Solve the resulting equation in one variable.
4. Substitute back to find the other variable.

When to use: When one of the equations already has a variable with coefficient 1 or $-1$, making the expression easy.

The elimination method involves multiplying the equations by suitable numbers so that the coefficients of one variable become equal (or negatives of each other), then adding or subtracting to eliminate that variable.

Steps:
1. Multiply one or both equations by suitable constants so that the coefficients of one variable become equal in magnitude.
2. Add or subtract the equations to eliminate that variable.
3. Solve for the remaining variable.
4. Substitute back to find the other variable.

When to use: This is the most versatile and commonly used method. It works well in almost all situations and is especially efficient when both equations have integer coefficients.

The cross-multiplication method provides a direct formula for the solution. For the system:

The solution is given by:

Memory trick: Write the coefficients in a specific pattern:

Each $2 \times 2$ determinant follows the pattern: top-left $\times$ bottom-right $-$ top-right $\times$ bottom-left.

When to use: When you want a formulaic approach. Especially useful when coefficients are messy or when you need both $x$ and $y$ simultaneously.

Some equations are not linear in $x$ and $y$ directly, but can be reduced to linear form by a suitable substitution. These are popular as 4-5 mark board exam questions.

Common types include equations with $\frac{1}{x}$ and $\frac{1}{y}$, or $\frac{a}{x+y}$ and $\frac{b}{x-y}$.

Word problems are the most common 4-5 mark questions from this chapter. The key skill is translating English sentences into mathematical equations.

General strategy:
1. Read the problem carefully. Identify what's unknown — assign variables ($x$ and $y$).
2. Translate each condition into an equation.
3. Solve using any method.
4. State the answer in context (not just "$x = 5$").

1. Sign Errors in Elimination:
* Mistake: Subtracting incorrectly, especially when both terms have the same sign.
* Fix: When subtracting, change ALL signs in the second equation, then add. Write it out explicitly.

2. Wrong Consistency Condition:
* Mistake: Confusing the conditions for no solution and infinitely many solutions.
* Fix: Remember: parallel lines ($\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$) = no solution. Coincident lines ($\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$) = infinite solutions.

3. Forgetting to Substitute Back:
* Mistake: Finding $x$ but forgetting to find $y$, or vice versa.
* Fix: Always substitute back to find both variables. A complete answer has values for both $x$ and $y$.

4. Cross-Multiplication Sign Errors:
* Mistake: Miscalculating the determinants in the cross-multiplication formula.
* Fix: Write the coefficients in the standard form ($a_1x + b_1y + c_1 = 0$) first. Be careful with signs when $c_1$ and $c_2$ are the constant terms moved to the left side.

5. Not Converting to Standard Form:
* Mistake: Applying formulas when the equations are not in the correct form.
* Fix: Always rearrange equations into $ax + by + c = 0$ or $ax + by = c$ form before applying any method.

6. Incorrect Variable Assignment in Word Problems:
* Mistake: Assigning variables without clearly defining them.
* Fix: Always write "Let $x = ...$" and "Let $y = ...$" at the start. State the answer in words at the end.

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

Cross-Multiplication Formula:

Substitution: Express one variable from one equation, plug into the other.

Elimination: Make coefficients equal, add/subtract.

Reduction: Let $u = \frac{1}{x}$, $v = \frac{1}{y}$ (or similar), solve the linear system, then convert back.

Pair of Linear Equations rewards practice — the more word problems you solve, the better you get at spotting patterns and setting up equations quickly.

Here's how SparkEd helps:

* Method-Wise Practice: Our Pair of Linear Equations page lets you practice substitution, elimination, and cross-multiplication problems separately before mixing them up.

* AI Math Solver: Paste any word problem into the AI Solver to see how to translate it into equations and solve step by step. It's perfect for learning the "setup" skill.

* AI Coach: Identifies whether your mistakes are in equation setup, algebraic manipulation, or final computation. Gives you targeted practice where you need it most.

* Algebra Connections: This chapter connects to Polynomials (for understanding equations) and Quadratic Equations (for the next level of equation solving).

Visit sparkedmaths.com and start solving!