Pair of Linear Equations in Two Variables (Class 10): Notes और MCQ Practice

आसान भाषा में

दो variables (x और y) वाली ऐसी दो equations जो straight lines को represent करती हैं। इन्हें साथ में solve करने का मतलब है वह common point (x, y) ढूंढना जहाँ ये दोनों lines graph पर एक-दूसरे से मिलती हैं।

Important Formulas & Conditions

Standard Form:

a1x + b1y + c1 = 0

a2x + b2y + c2 = 0

Ratio Comparison (Direct Questions in Board Exam):

1. Intersecting Lines:

a1/a2 ≠ b1/b2 → Unique Solution (Consistent)

1. Coincident Lines (एक के ऊपर एक):

a1/a2 = b1/b2 = c1/c2 → Infinitely Many Solutions (Consistent / Dependent)

1. Parallel Lines:

a1/a2 = b1/b2 ≠ c1/c2 → No Solution (Inconsistent)

Solving Methods:

• Substitution Method: एक equation से x की value निकाल कर दूसरी equation में substitute करना।
• Elimination Method: किसी एक variable के coefficients equal बनाकर उसे cancel (eliminate) करना।

2 Memory Tricks

1. "Train की पटरी" (Parallel Lines): जैसे रेल की पटरियां कभी नहीं मिलतीं, वैसे ही Parallel lines का कोई meeting point नहीं होता → यानी No Solution!
2. "पहला अलग तो अनोखा" (Unique): अगर पहला ratio ही equal नहीं है (a1/a2 ≠ b1/b2), तो आगे c1/c2 चेक ही मत करो, सीधे Unique Solution लिख दो।

2 Common Board Exam Mistakes

1. Signs (+ / −) की गलती: Ratios निकालते समय equations को standard form में लाएं बिना constant terms (c1, c2) का sign गलत ले लेना। हमेशा sign साथ रखें!
2. Elimination में अधूरा Multiplication: किसी equation को multiply करते समय students LHS को तो multiply कर देते हैं, लेकिन RHS (equal to के बाद वाले नंबर) को multiply करना भूल जाते हैं।

Pair of Linear Equations in Two Variables

1. Chapter in Short

In Class 9, you learned about a single linear equation in two variables (such as 2x + 3y = 5), which has infinitely many solutions. In Class 10, we study a pair (system) of two linear equations in the same two variables.

Finding a "solution" means finding a pair of values for x and y that makes both equations true at the same time. Geometrically, each equation represents a straight line on a graph. The chapter teaches us how to find whether these lines meet, overlap, or run parallel, and provides algebraic methods to calculate the exact values of x and y.

2. Key Concepts Explained Simply

What is a Linear Equation in Two Variables?

Imagine you go to a stationery shop and buy 2 pens and 3 notebooks for Rs 40.

If the price of one pen is 'x' and one notebook is 'y', this situation can be written as:

2x + 3y = 40

Since the highest power of both x and y is 1, it is called a linear equation. When plotted on graph paper, it always forms a straight line.

What Happens When You Have Two Lines?

When two straight lines are drawn on the same flat piece of paper, only three situations are possible:

1. Intersecting Lines (Crossing paths):

• Like a road crossing or an "X".
• The lines cross at exactly one point.
• Therefore, there is exactly one (unique) solution.
• The system is called consistent.

1. Parallel Lines (Railway tracks):

• Like railway tracks, the lines never meet, no matter how far they extend.
• Since they never touch, there is no common point.
• Therefore, there is no solution.
• The system is called inconsistent.

1. Coincident Lines (Overlapping paths):

• One line lies directly on top of the other line.
• Every point on the line is shared.
• Therefore, there are infinitely many solutions.
• The system is called dependent (consistent).

Algebraic Methods to Solve Equations

Instead of drawing graphs every time, we use two reliable algebraic methods:

1. Substitution Method:

• Pick one equation and express one variable in terms of the other (for example, write x = 5 - 2y).
• Substitute this expression into the other equation to find the value of y.
• Put that value back to get x.

1. Elimination Method:

• Multiply one or both equations by numbers so that the coefficients of either x or y become equal.
• Add or subtract the two equations to "eliminate" (remove) that variable.
• Solve the simple one-variable equation left behind, then find the second variable.

3. Important Definitions, Formulas, and Facts

General Form

A pair of linear equations in two variables is written as:

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

(where a1, b1, c1, a2, b2, c2 are real numbers, and a1² + b1² ≠ 0, a2² + b2² ≠ 0)

Comparing the Ratios (Without Drawing Graphs)

Compare the coefficients: a1/a2, b1/b2, and c1/c2:

1. If a1/a2 ≠ b1/b2:

• Graphical representation: Intersecting lines
• Algebraic interpretation: Exactly one unique solution
• Consistency: Consistent

1. If a1/a2 = b1/b2 = c1/c2:

• Graphical representation: Coincident lines (overlapping)
• Algebraic interpretation: Infinitely many solutions
• Consistency: Consistent (dependent)

1. If a1/a2 = b1/b2 ≠ c1/c2:

• Graphical representation: Parallel lines
• Algebraic interpretation: No solution
• Consistency: Inconsistent

4. Solved Examples (Board-Exam Style)

Example 1: Checking Nature of Solutions Using Ratios

Question: Check whether the pair of equations 2x + 3y - 9 = 0 and 4x + 6y - 18 = 0 has a unique solution, infinitely many solutions, or no solution.

Solution:

• Here, a1 = 2, b1 = 3, c1 = -9
• a2 = 4, b2 = 6, c2 = -18

Find the ratios:

• a1/a2 = 2/4 = 1/2
• b1/b2 = 3/6 = 1/2
• c1/c2 = (-9)/(-18) = 1/2

Since a1/a2 = b1/b2 = c1/c2 = 1/2, the lines are coincident.

Answer: The pair has infinitely many solutions (consistent).

Example 2: Solving by Substitution Method

Question: Solve the following pair of equations using the substitution method:

1. x + y = 14
2. x - y = 4

Solution:

• From equation (2), express x in terms of y:

x = 4 + y ...(Equation 3)

• Substitute this value of x into equation (1):

(4 + y) + y = 14

4 + 2y = 14

2y = 14 - 4

2y = 10

y = 10 ÷ 2 = 5

• Substitute y = 5 into equation (3):

x = 4 + 5 = 9

Answer: x = 9, y = 5

Example 3: Solving by Elimination Method

Question: Solve the following pair of equations using the elimination method:

1. 3x + 4y = 10
2. 2x - 2y = 2

Solution:

• Make the coefficients of y equal. Multiply equation (2) by 2:

2 × (2x - 2y = 2) => 4x - 4y = 4 ...(Equation 3)

• Now add equation (1) and equation (3) to eliminate y:

(3x + 4y) + (4x - 4y) = 10 + 4

7x = 14

x = 14 ÷ 7 = 2

• Substitute x = 2 into equation (1):

3(2) + 4y = 10

6 + 4y = 10

4y = 10 - 6

4y = 4

y = 1

Answer: x = 2, y = 1

Example 4: Word Problem (Ages)

Question: Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu now?

Solution:

• Let the present age of Nuri be 'x' years, and the present age of Sonu be 'y' years.

Condition 1 (Five years ago):

• Nuri's age = x - 5
• Sonu's age = y - 5
• Equation: (x - 5) = 3 × (y - 5)

x - 5 = 3y - 15

x - 3y = -15 + 5

x - 3y = -10 ...(Equation 1)

Condition 2 (Ten years later):

• Nuri's age = x + 10
• Sonu's age = y + 10
• Equation: (x + 10) = 2 × (y + 10)

x + 10 = 2y + 20

x - 2y = 20 - 10

x - 2y = 10 ...(Equation 2)

Solving by elimination:

• Subtract equation (1) from equation (2):

(x - 2y) - (x - 3y) = 10 - (-10)

-2y + 3y = 10 + 10

y = 20

• Put y = 20 in equation (2):

x - 2(20) = 10

x - 40 = 10

x = 50

Answer: Present age of Nuri is 50 years, and present age of Sonu is 20 years.

5. Common Mistakes and Quick Revision Points

Common Mistakes to Avoid:

1. Sign Errors with Constants (c1 and c2):

Before comparing ratios, make sure both equations are in the same form. Either write both as ax + by + c = 0, or both as ax + by = c. If one equation has the constant on the left and the other on the right, signs will get mixed up.

1. Sign Errors when Subtracting Equations:

In the elimination method, when you subtract one equation from another, remember to change the sign of every single term in the lower equation.

1. Stopping Halfway:

After finding the value of x, students often forget to calculate the value of y. Always ensure you have found both values.

1. Word Problem Misinterpretation:

"5 years ago" means subtract 5 from both ages. "10 years hence / later" means add 10 to both ages. Never apply the age multiplier only to the present age.

Quick Revision Points:

• Unique solution: a1/a2 ≠ b1/b2 (Lines intersect at 1 point; Consistent).
• Infinitely many solutions: a1/a2 = b1/b2 = c1/c2 (Lines coincide; Consistent).
• No solution: a1/a2 = b1/b2 ≠ c1/c2 (Lines are parallel; Inconsistent).
• Self-Check Trick: Always plug your final values of x and y back into both original equations. If both equations balance, your answer is 100% correct!