Pair of Linear Equations in Two Variables Class 10 Notes

This guide covers Pair of Linear Equations in Two Variables Class 10, chapter 3 of NCERT Class 10 Mathematics. You will learn how to tell from a graph or from coefficient ratios whether a pair has one, many or no solutions, and how to solve it by substitution and elimination. Worked word problems, important questions and common mistakes follow.

What a pair of linear equations is

A pair of linear equations in two variables x and y has the general form:

a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0

Here all the a, b and c values are real numbers, with a₁² + b₁² ≠ 0 and a₂² + b₂² ≠ 0. In simple words, x and y do not both have a zero coefficient in either equation. They are called linear because each graph is a straight line.

Example from daily life: Akhila spends 20 rupees at a fair on Giant Wheel rides (x) and Hoopla games (y). A ride costs 3 rupees and Hoopla costs 4 rupees, so 3x + 4y = 20. She played Hoopla half as many times as she rode the wheel, so y = (1/2)x. These two equations form a pair.

Graphical method and consistency

Draw both lines on the same graph. The solution is the point where they meet. Three things can happen:

A pair is consistent if it has at least one solution, and inconsistent if it has none.

The ratio test (no graph needed)

Compare a₁/a₂, b₁/b₂ and c₁/c₂:

These ratios only make sense when the denominators are not zero. If a coefficient is zero, compare the equations directly; never divide by zero. For example, x = 2 and y = 3 still have one solution even though some ratios are undefined.

The substitution method

This method is handy when a variable has coefficient 1 or −1.

1. Make one variable the subject of one equation.
2. Substitute that expression into the other equation.
3. Solve the one-variable equation.
4. Put the value back to find the other variable.

Example: 7x − 15y = 2 and x + 2y = 3.

• From the second equation, x = 3 − 2y.
• Substitute: 7(3 − 2y) − 15y = 2, so 21 − 14y − 15y = 2.
• 21 − 29y = 2, so −29y = −19 and y = 19/29.
• x = 3 − 2(19/29) = (87 − 38)/29 = 49/29.
• Check: 7(49/29) − 15(19/29) = 58/29 = 2, and 49/29 + 38/29 = 87/29 = 3.

If the variables cancel and leave a true statement like 0 = 0 or 18 = 18, there are infinitely many solutions. If they leave a false statement like −4 = 0, there is no solution.

The elimination method

1. Multiply one or both equations by non-zero numbers so that one variable has numerically equal coefficients.
2. If the signs are opposite, add the equations. If the signs are the same, subtract.
3. Solve for the remaining variable, then substitute back.

Example: 9x − 4y = 2000 and 7x − 3y = 2000.

• Multiply the first by 3: 27x − 12y = 6000. Multiply the second by 4: 28x − 12y = 8000.
• Subtract the first new equation from the second: x = 2000.
• Substitute: 9(2000) − 4y = 2000, so 18000 − 4y = 2000, 4y = 16000 and y = 4000.
• Check: 9(2000) − 4(4000) = 2000 and 7(2000) − 3(4000) = 2000.

Elimination is often faster when coefficients are large or fractional.

Word problems: setting up the equations

• Ages: if someone is x years old now, they were (x − n) years old n years ago and will be (x + n) years old n years from now.
• Two-digit numbers: with tens digit x and units digit y, the number is 10x + y. Reversed, it is 10y + x. Never write the number as x × y.
• Fixed charge plus rate: Total cost = Fixed charge + (Rate per unit × Number of units). Two different situations give two equations.
• Fractions: call the fraction x/y, apply the change described, set it equal to the new value, and cross-multiply.

Remember this

• One solution: a₁/a₂ ≠ b₁/b₂. Infinitely many: all three ratios equal. None: a₁/a₂ = b₁/b₂ ≠ c₁/c₂.
• Consistent means at least one solution. Parallel lines mean inconsistent.
• Substitution: true leftover statement means infinitely many solutions; false means none.
• Always check your answer in both original equations.

Important questions with answers

1. If a₁/a₂ = 1/2, b₁/b₂ = 1/2 and c₁/c₂ = 5/6, how many solutions are there? a₁/a₂ = b₁/b₂ ≠ c₁/c₂, so the lines are parallel and there is no solution.

2. Without drawing, decide whether x + 2y − 4 = 0 and 2x + 4y − 12 = 0 are consistent. a₁/a₂ = 1/2, b₁/b₂ = 2/4 = 1/2, c₁/c₂ = −4/−12 = 1/3. The first two are equal but differ from the third, so the lines are parallel. The pair is inconsistent.

3. Solve Akhila's pair: 3x + 4y = 20 and y = (1/2)x. Substitute: 3x + 4(x/2) = 20, so 3x + 2x = 20 and x = 4. Then y = 2. She took 4 rides and played Hoopla 2 times. Check: 3(4) + 4(2) = 20.

4. Solve 2x + 3y = 10 and 2x − 5y = 2 by elimination. Subtract the second from the first: 8y = 8, so y = 1. Then 2x + 3 = 10, so x = 7/2. Check: 2(7/2) − 5(1) = 7 − 5 = 2.

5. The sum of a two-digit number and the number formed by reversing its digits is 66. The digits differ by 2, and the tens digit is larger. Find the number. (10x + y) + (10y + x) = 66 gives 11x + 11y = 66, so x + y = 6. Also x − y = 2. Adding: 2x = 8, x = 4, y = 2. The number is 42. (If the units digit were larger, the number would be 24.)

6. A father is 3 times as old as his son, and their ages add up to 48. Find their ages. Son = x, father = 3x. Then 4x = 48, x = 12. Son is 12, father is 36.

7. A fraction becomes 1/3 when 1 is subtracted from the numerator, and 1/4 when 8 is added to the denominator. Find it. (x − 1)/y = 1/3 gives 3x − y = 3. x/(y + 8) = 1/4 gives 4x − y = 8. Subtracting: x = 5, so y = 12. The fraction is 5/12. Check: 4/12 = 1/3 and 5/20 = 1/4.

8. What does a result like 0 = 0 mean during substitution? The equations are dependent (coincident lines), so there are infinitely many solutions.

Common mistakes to avoid

• Dividing by a zero coefficient in the ratio test.
• Mixing up the conditions for parallel and coincident lines. Coincident needs all three ratios equal.
• Adding equations when the coefficients have the same sign. Same signs: subtract.
• Writing a two-digit number as xy instead of 10x + y.
• Skipping the final check in both equations.

Try each method on the same pair to see which feels quicker for you, and study this chapter with Joy for more practice.