Class 10 Linear Equations Notes

Exam Weightage & Blueprint

Total: 5-6 Marks

This chapter falls under Unit II: Algebra (20 marks total). As per the latest syllabus, focus is on: graphical solution of pairs of equations, consistency conditions, and algebraic methods (substitution and elimination) for solving situational problems.

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

Geometric Behavior: Intersecting ($a_1/a_2 \neq b_1/b_2$), Coincident ($=b_1/b_2 = c_1/c_2$), Parallel ($=b_1/b_2 \neq c_1/c_2$)

Algebraic Methods for Solving

1. Substitution Method

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

2. Elimination Method

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

⚡ Board Special Shortcut: Interchanged Large Coefficients

For equations of the form: $ax + by = c$ and $bx + ay = d$ (e.g., $152x - 378y = -74$ and $-378x + 152y = -604$):

Step 1 (Add): Add both equations → $(a+b)x + (a+b)y = c+d \implies \mathbf{x + y = p}$
Step 2 (Subtract): Subtract both equations → $(a-b)x - (a-b)y = c-d \implies \mathbf{x - y = q}$
Step 3 (Solve): Simply add and subtract these two tiny equations to find $x$ and $y$ in seconds!

🚀 Word Problem Equation Models (CBSE Blueprint):

Two-digit number: Number $= 10x + y$, Reversed $= 10y + x$.
Fractions: Fraction $= \frac{x}{y}$ (where $x =$ numerator, $y =$ denominator).
Fixed + Variable charges: Total Cost $= \text{Fixed Charge } (x) + (\text{days/km } \times y)$.
Upstream & Downstream: Let boat speed $= x\text{ km/h}$, stream speed $= y\text{ km/h}$.
$\text{Upstream Speed } = x - y$, $\text{Downstream Speed } = x + y$. $\text{Time } = \frac{\text{Distance}}{\text{Speed}}$.

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.

📈 Graphical Method of Solution

Each linear equation $ax + by + c = 0$ represents a straight line on the graph. The solution to the pair is the point of intersection of the two lines.

Steps to Solve Graphically:

Express each equation as $y = \frac{-ax - c}{b}$
Find 2–3 ordered pairs $(x, y)$ satisfying each equation
Plot points and draw both lines on the same graph
The intersection point (if any) is the solution

✅ Intersecting Lines
One common point → Unique Solution
Consistent pair

♾️ Coincident Lines
Infinite points → Infinite Solutions
Dependent/Consistent

⚫ Parallel Lines
No common point → No Solution → Inconsistent pair

⚠️ Board Tip: In graphical questions, always find at least 3 points per line for accuracy, and clearly label the intersection point as the solution.

📝 More Solved Board Questions

Q3. Solve by Elimination: $3x + 2y = 11$ and $2x + 3y = 4$. 3 Marks

Sol. Multiply Eq 1 by 3 and Eq 2 by 2:

$9x + 6y = 33$ ...(3)

$4x + 6y = 8$ ...(4)

Subtract (4) from (3): $5x = 25 \Rightarrow x = 5$

Substitute in Eq 1: $15 + 2y = 11 \Rightarrow y = -2$

Answer: $x = 5, y = -2$

Q4. Solve for $x$ and $y$: $152x - 378y = -74$ and $-378x + 152y = -604$. 3 Marks (Frequent Standard Paper)

Sol. Coefficients of $x$ and $y$ are interchanged. Use the Add & Subtract method.

Adding both equations:
$(152 - 378)x + (-378 + 152)y = -74 - 604$
$\Rightarrow -226x - 226y = -678 \implies -226(x + y) = -678 \implies \mathbf{x + y = 3}$ ...(3)

Subtracting second from first:
$(152 - (-378))x + (-378 - 152)y = -74 - (-604)$
$\Rightarrow 530x - 530y = 530 \implies \mathbf{x - y = 1}$ ...(4)

Adding (3) and (4): $2x = 4 \Rightarrow \mathbf{x = 2}$.
Subtracting (4) from (3): $2y = 2 \Rightarrow \mathbf{y = 1}$.
Answer: $x = 2, y = 1$

Q5. A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. Determine the speed of the stream and that of the boat in still water. 5 Marks (Long Answer)

Sol. Let speed of boat in still water $= x\text{ km/h}$ and speed of stream $= y\text{ km/h}$.

Speed upstream $= (x - y)\text{ km/h}$, Speed downstream $= (x + y)\text{ km/h}$.

$\frac{30}{x-y} + \frac{44}{x+y} = 10$ ...(1) and $\frac{40}{x-y} + \frac{55}{x+y} = 13$ ...(2)

Let $\frac{1}{x-y} = u$ and $\frac{1}{x+y} = v$:

$30u + 44v = 10 \Rightarrow 15u + 22v = 5$ ...(3)
$40u + 55v = 13$ ...(4)

Multiplying (3) by 8 and (4) by 3:
$120u + 176v = 40$
$120u + 165v = 39$
Subtracting gives $11v = 1 \Rightarrow v = \frac{1}{11} \implies x + y = 11$.
Substituting $v$ into (3): $15u + 2 = 5 \Rightarrow 15u = 3 \Rightarrow u = \frac{1}{5} \implies x - y = 5$.

Adding $x+y=11$ and $x-y=5$: $2x = 16 \Rightarrow \mathbf{x = 8\text{ km/h}}$.
Then $y = 11 - 8 = \mathbf{3\text{ km/h}}$.
Speed of boat = 8 km/h, Speed of stream = 3 km/h.

Q6. Find the value of $k$ for which equations $2x + 3y = 7$ and $(k-1)x + (k+2)y = 3k$ have infinitely many solutions. 3 Marks

Sol. For infinitely many solutions: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

$\frac{2}{k-1} = \frac{3}{k+2} = \frac{7}{3k}$

From first two: $2(k+2) = 3(k-1) \Rightarrow 2k + 4 = 3k - 3 \Rightarrow k = 7$

Verify with third ratio: $\frac{7}{21} = \frac{1}{3}$ and $\frac{2}{6} = \frac{1}{3}$ ✅

$k = 7$

🎯 Board Pattern (2018–2025): Word problems (ages, money, fractions, upstream/downstream) appear almost every year as 3-mark or 5-mark questions. Setting up equations with clear “Let statements” guarantees partial marks even if an arithmetic slip happens later.

📋 Board Revision Checklist

✅ General form: $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$
✅ $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$ → Intersecting → Unique solution (Consistent)
✅ $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ → Coincident → Infinite solutions (Consistent/Dependent)
✅ $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$ → Parallel → No solution (Inconsistent)
✅ Interchanged coefficients ($ax+by=c, bx+ay=d$) → Add once ($x+y=p$), Subtract once ($x-y=q$)
✅ Upstream speed $= (x - y)$, Downstream speed $= (x + y)$
✅ Substitution: express one variable, substitute into other equation
✅ Elimination: multiply to equate coefficients, then add/subtract
✅ Two-digit number: tens digit $x$, units digit $y$ → number $= 10x + y$, reverse $= 10y + x$
✅ Always verify answer by substituting back in BOTH equations

💡 Exam Tip:
For the consistency MCQ, memorize: “Different ratios → Unique → Consistent; Same all three → Infinite; Same first two, different third → No solution.”

Concept Mastery Quiz 🎯

Test your readiness for the board exam.

1. If $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$, the pair of equations has:

2. The graphical representation of two coincident lines means:

3. For $x + 2y = 5$ and $3x + 6y = 15$, the pair is:

4. In a two-digit number, if tens digit is $x$ and units digit is $y$, the number reversed is:

5. The value of $k$ for which $x + ky = 2$ and $2x + 4y = 6$ has no solution is:

Chapter 3: Pair of Linear Equations in Two Variables

Overview