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Chapter 3: Pair of Linear Equations in Two Variables

Board Exam Focused Notes, Methods, and PYQs

Exam Weightage & Blueprint

Total: 5-6 Marks

This chapter is part of the Algebra Unit (20 Marks Total). Focus on algebraic methods and consistency conditions.

Question Type Marks Frequency Focus Topic
MCQ 1 High Consistency Conditions ($a_1/a_2$...)
Short Answer 2 or 3 Medium Substitution/Elimination Method
Word Problem 3 or 5 High Ages, Digits, Fractions

⏰ Last 24-Hour Checklist

  • General Form: $a_1x + b_1y + c_1 = 0$.
  • Consistency Table: Memorize unique, no, and infinite solution conditions.
  • Elimination Method: Equate coefficients and subtract.
  • Substitution Method: Express $x$ in terms of $y$.
  • Digit Problems: Number $= 10x + y$. Reverse $= 10y + x$.
  • Fraction Problems: Assume fraction as $x/y$.

Consistency & Graph Nature ★★★★★

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

Ratio Condition Graphical Representation Algebraic Interpretation Consistency
$\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$ Intersecting Lines Unique Solution Consistent
$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ Coincident Lines Infinitely Many Solutions Dependent
$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$ Parallel Lines No Solution Inconsistent

Algebraic Methods for Solving

1. Substitution Method

  1. Find value of one variable ($y$) in terms of other ($x$) from Eq 1.
  2. Substitute this into Eq 2 to get equation in one variable.
  3. Solve for $x$.
  4. Put $x$ back in Step 1 to find $y$.

2. Elimination Method

  1. Multiply equations by constants to make coefficients of one variable equal.
  2. Add or Subtract equations to eliminate that variable.
  3. Solve for the remaining variable.
  4. Substitute back to find the eliminated variable.

Solved Examples (Board Marking Scheme)

Q1. The sum of a two-digit number and the number obtained by reversing the digits is 66. If the digits differ by 2, find the number. (3 Marks)

Step 1: Forming Equations 1 Mark

Let tens digit be $x$ and units digit be $y$. Number $= 10x + y$.

Reverse Number $= 10y + x$.

Case 1: Sum is 66 $\Rightarrow (10x+y) + (10y+x) = 66 \Rightarrow 11(x+y)=66 \Rightarrow x+y=6$ ...(1)

Case 2: Digits differ by 2 $\Rightarrow x-y=2$ ...(2) OR $y-x=2$ ...(3)

Step 2: Solving 1 Mark

Add (1) and (2): $2x = 8 \Rightarrow x=4$. Then $y=2$. Number is 42.

Add (1) and (3): $2y = 8 \Rightarrow y=4$. Then $x=2$. Number is 24.

Step 3: Conclusion 1 Mark

There are two such numbers: 42 and 24.

Q2. Check if $x-2y=0$ and $3x+4y-20=0$ are consistent. (2 Marks)

Step 1: Compare Ratios 1 Mark

$a_1=1, b_1=-2, c_1=0$

$a_2=3, b_2=4, c_2=-20$

$\frac{a_1}{a_2} = \frac{1}{3}$ and $\frac{b_1}{b_2} = \frac{-2}{4} = -\frac{1}{2}$.

Step 2: Conclusion 1 Mark

Since $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$, the lines intersect at one point.

Therefore, the pair of equations is consistent with a unique solution.

Previous Year Questions (PYQs)

2023 (1 Mark): Find the value of $k$ for which equations $x+2y=3$ and $5x+ky+7=0$ are inconsistent.
Hint: Condition for inconsistent (parallel): $\frac{1}{5} = \frac{2}{k} \neq \frac{3}{-7} \Rightarrow k=10$.
2020 (3 Marks): 5 pencils and 7 pens cost ₹50, whereas 7 pencils and 5 pens cost ₹46. Find cost of one pencil and one pen.
Ans: Pencil (x)=₹3, Pen (y)=₹5. (Solve $5x+7y=50, 7x+5y=46$).
2019 (3 Marks): A fraction becomes 9/11 if 2 is added to both numerator and denominator. If 3 is added to both, it becomes 5/6. Find the fraction.
Ans: Fraction is 7/9. (Equations: $11x-9y=-4$ and $6x-5y=-3$).

Exam Strategy & Mistake Bank

Mistake Bank 🚨

Sign Errors: When standard form is $ax+by=c$, $c_1/c_2$ ratio sign can be tricky. Always bring to $ax+by+c=0$.
Digit Problem: Writing number as $xy$ instead of $10x+y$.
Unit Conversions: In speed/distance problems, ensure km/h and mins are converted correctly.

Scoring Tips 🏆

Let Statement: Always start word problems with "Let the number be x..."
Verify: If time permits, put your values of x and y back into the question to check.
Nature of Lines: Memorize the table. 1 Mark is guaranteed from there.

Self-Assessment Mock Test (10 Marks)

Q1 (1M): Write the condition for a pair of linear equations to have infinitely many solutions.

Q2 (2M): Solve for x and y: $x+y=14$, $x-y=4$.

Q3 (3M): Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. Find their ages.

Q4 (4M): Meena went to bank to withdraw ₹2000. She received only ₹50 and ₹100 notes. Total notes are 25. Find number of notes of each type.

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