The chapter on pairs of linear equations teaches several ways to solve the same problem, and students often treat them as interchangeable. They are not. Each method has a purpose, and choosing the right one saves time and avoids errors. The graphical method shows you what the answer looks like. The algebraic methods give you the answer exactly.
This guide explains what each method does best, solves one problem by every method so that you can compare them, and ends with a simple way to choose. If you need the basics first, read these notes on the pair of linear equations in two variables, the graphical method and the algebraic methods.
What a Solution Actually Means
A pair of linear equations in two variables describes two straight lines. A solution is a point that lies on both lines. That one idea gives three possible cases, and every method is a way of finding out which case you are in.
| What the lines do | Number of solutions | Name | Condition on the coefficients |
| Intersect at one point | Exactly one | Consistent | a₁/a₂ ≠ b₁/b₂ |
| Coincide (lie on top of each other) | Infinitely many | Consistent (dependent) | a₁/a₂ = b₁/b₂ = c₁/c₂ |
| Are parallel | None | Inconsistent | a₁/a₂ = b₁/b₂ ≠ c₁/c₂ |
The equations are taken in the form a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0. Comparing the ratios of the coefficients tells you the case without solving anything, and it is often the quickest way to answer an “is it consistent?” question.
One Problem, Four Ways
Solve the pair 2x + 3y = 13 and 5x − 2y = 4.
Graphical method
Make a table of points for each line and plot them.
| Line | Point 1 | Point 2 | Point 3 |
| 2x + 3y = 13 | (2, 3) | (5, 1) | (−1, 5) |
| 5x − 2y = 4 | (2, 3) | (0, −2) | (4, 8) |
Both lines pass through (2, 3), so that is where they intersect. Answer: x = 2, y = 3. The graph shows the answer to the eye, but its accuracy depends on how carefully you plot, so it is best when the answer is a whole number.
Substitution
Solve one equation for one variable and substitute into the other. Here neither variable has a coefficient of 1, so the working gets awkward with fractions. Substitution is best when one variable has a coefficient of 1 or −1, as in the pair x − y = 2 and 3x + 2y = 11. From the first, x = y + 2. Substituting, 3(y + 2) + 2y = 11, so 5y = 5 and y = 1, giving x = 3.
Elimination
Make the coefficients of one variable equal in size, and add or subtract.
- Multiply the first equation by 2: 4x + 6y = 26.
- Multiply the second by 3: 15x − 6y = 12.
- Add: 19x = 38, so x = 2.
- Substitute: 2(2) + 3y = 13, so 3y = 9 and y = 3.
Elimination is the most dependable all-round method, especially when the coefficients are awkward.
Cross-multiplication
Write the equations as 2x + 3y − 13 = 0 and 5x − 2y − 4 = 0, so a₁ = 2, b₁ = 3, c₁ = −13, a₂ = 5, b₂ = −2, c₂ = −4. The formulas are
x = (b₁c₂ − b₂c₁) ÷ (a₁b₂ − a₂b₁), y = (c₁a₂ − c₂a₁) ÷ (a₁b₂ − a₂b₁).
The denominator is 2(−2) − 5(3) = −19. Then x = (3(−4) − (−2)(−13)) ÷ (−19) = (−12 − 26) ÷ (−19) = 2, and y = ((−13)(5) − (−4)(2)) ÷ (−19) = (−65 + 8) ÷ (−19) = 3. The method is fast once the signs are handled with care, and the sign handling is where the marks are lost.
How to Choose
| The situation | Best method | Why |
| The question says “graphically” or asks you to draw | Graphical | It is required, and the picture is the answer |
| You only need to know whether a solution exists | Compare the ratios of coefficients | No solving is needed |
| One variable has a coefficient of 1 or −1 | Substitution | Simple, with no fractions |
| Coefficients are similar or easy to match | Elimination | Quick and tidy |
| The coefficients are awkward and you are confident with signs | Cross-multiplication | A direct formula, but sign-sensitive |
| You want to check an answer | Substitute back into both equations | Catches arithmetic slips |
Three Special Cases
Coincident lines. If 2 pencils and 3 erasers cost ₹9, and 4 pencils and 6 erasers cost ₹18, the equations are 2x + 3y = 9 and 4x + 6y = 18. The ratios are 2/4 = 3/6 = 9/18 = 1/2, so the lines coincide and there are infinitely many solutions. The second purchase is just double the first, so it gives no new information.
Parallel lines. For x + 2y = 4 and 2x + 4y = 12, the ratios are 1/2 = 2/4 ≠ 4/12. The lines are parallel, so there is no solution. In algebra, elimination produces an impossible statement such as 0 = 4.
Finding an unknown constant. A common board question asks for the value of k for which a pair has no solution, or infinitely many. Set up the ratio condition and solve for k.
From Words to Equations
Most board questions arrive as stories, and the real work is writing the equations.
- Name the unknowns clearly, for example: let the cost of one pencil be ₹x and of one eraser be ₹y.
- Translate each sentence into one equation. The number of equations should equal the number of unknowns.
- Solve with the most convenient method.
- Answer in words, with units, and check that the answer makes sense, for example that a price is not negative.
Typical stories involve ages, speeds of boats in still water and in a stream, fractions with a changed numerator and denominator, and costs of two sets of items.
Common Slips
- Writing the equations in different forms and then reading the wrong coefficients.
- Mishandling the sign of c when moving to the form ax + by + c = 0.
- Not multiplying every term of an equation when preparing for elimination.
- Concluding “no solution” from a sign error, when the lines actually intersect.
- Giving x and y without checking them in both equations.
To see how the habits in this chapter carry over to equations of a higher degree, read why a quadratic equation has two roots and common mistakes in the quadratic formula.
Linear Equations in Your Syllabus
Pair of Linear Equations in Two Variables is a chapter of the Class 10 Maths book in both the CBSE and the Karnataka SSLC textbooks. Practise with these chapter MCQs and these important questions for board exams. The ideas return in Class 11 and 12, in coordinate geometry and in matrices and determinants, which matter for JEE Main and KCET. Confirm the current portions with your own board.
Frequently Asked Questions
Q1. What are the three cases for a pair of linear equations?
The lines can intersect at one point (a unique solution), coincide (infinitely many solutions) or be parallel (no solution).
Q2. When should I use the graphical method?
When the question asks for it, or when you want to see how the two lines relate. For exact answers, especially with fractions, the algebraic methods are more reliable.
Q3. Which algebraic method is best?
Elimination is the most dependable all-round method. Substitution suits cases where one variable has a coefficient of 1, and cross-multiplication is a quick formula when you are careful with signs.
Q4. How can I tell, without solving, whether a pair has no solution?
Compare the ratios. If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the lines are parallel and there is no solution.
Q5. How do I check my answer?
Substitute the values of x and y into both original equations. They must satisfy both.
Match the Method to the Question
The methods are not rivals. A graph shows you the shape of the answer, the ratios tell you how many answers there are, and the algebraic methods deliver the exact values. Know what each one is for, and you will always have the quickest route in hand.
Get Social