Exercise 3.3

Similar to Q1(i)

Solve using elimination: x + y = 7 and 3x − 2y = 11

View Step-by-Step Solution

Step 1: Equate Coefficients

Multiply eq(1) by 2: 2x + 2y = 14 ... (3)
Keep eq(2): 3x − 2y = 11 ... (2)

Step 2: Add Equations

(2x + 3x) + (2y - 2y) = 14 + 11
5x = 25 → x = 5

Step 3: Substitute

Put x = 5 in eq(1): 5 + y = 7 → y = 2

Answer: x = 5, y = 2

Similar to Q1(ii)

Solve: 2x + 3y = 8 and 4x − y = 2

View Step-by-Step Solution

Step 1: Equate Coefficients

Multiply eq(1) by 2: 4x + 6y = 16 ... (3)
Keep eq(2): 4x − y = 2 ... (2)

Step 2: Subtract (3) - (2)

(4x - 4x) + (6y - (-y)) = 16 - 2
7y = 14 → y = 2

Step 3: Find x

Put y = 2 in eq(1): 2x + 3(2) = 8 → 2x = 2 → x = 1

Answer: x = 1, y = 2

Similar to Q1(iii)

Solve: 3x − 4y = 1 and 5x − 3y = 9

View Step-by-Step Solution

Step 1: Rearrange

Equations are already in form ax + by = c.

Step 2: Equate y

Multiply eq(1) by 3: 9x - 12y = 3 ...(3)
Multiply eq(2) by 4: 20x - 12y = 36 ...(4)

Step 3: Subtract (4) - (3)

(20x - 9x) = 36 - 3
11x = 33 → x = 3

Step 4: Find y

Put x = 3 in eq(1): 3(3) - 4y = 1 → 9 - 4y = 1 → -4y = -8 → y = 2

Answer: x = 3, y = 2

Similar to Q1(iv)

Solve: x/3 + y/4 = 2 and x − y/2 = 3

View Step-by-Step Solution

Step 1: Simplify Fractions

Eq(1) × 12: 4x + 3y = 24 ...(3)
Eq(2) × 2: 2x - y = 6 ...(4)

Step 2: Equate y

Multiply eq(4) by 3: 6x - 3y = 18 ...(5)
Keep eq(3): 4x + 3y = 24

Step 3: Add

10x = 42 → x = 4.2 (or 21/5)

Step 4: Find y

Put x = 4.2 in eq(2): 4.2 - y/2 = 3 → 1.2 = y/2 → y = 2.4 (or 12/5)

Answer: x = 21/5, y = 12/5

Similar to Q2(i)

If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes 1/2 if we add 3 to the denominator. What is the fraction?

View Step-by-Step Solution

Step 1: Form Equations

Let fraction be x/y.
Case 1: (x+1)/(y-1) = 1 → x + 1 = y - 1 → x - y = -2 ...(1)
Case 2: x/(y+3) = 1/2 → 2x = y + 3 → 2x - y = 3 ...(2)

Step 2: Eliminate y

Subtract (1) from (2):
(2x - x) - (y - y) = 3 - (-2)
x = 5

Step 3: Find y

Put x = 5 in (1): 5 - y = -2 → -y = -7 → y = 7

Fraction = 5/7

Similar to Q2(ii)

5 years ago, Father was 5 times as old as his son. 5 years later, Father will be 3 times as old as his son. Find their present ages.

View Step-by-Step Solution

Step 1: Form Equations

Let Father = x, Son = y.
5 years ago: (x-5) = 5(y-5) → x - 5y = -20 ...(1)
5 years later: (x+5) = 3(y+5) → x - 3y = 10 ...(2)

Step 2: Eliminate x

Subtract (1) from (2):
(-3y) - (-5y) = 10 - (-20)
2y = 30 → y = 15

Step 3: Find x

Put y = 15 in (2): x - 45 = 10 → x = 55

Father = 55 years, Son = 15 years

Similar to Q2(iii)

The sum of the digits of a two-digit number is 9. The number obtained by reversing the order of the digits exceeds the original number by 27. Find the number.

View Step-by-Step Solution

Step 1: Form Equations

Let number be 10x + y.
Sum of digits: x + y = 9 ...(1)
Reversed (10y + x) - Original (10x + y) = 27
9y - 9x = 27 (Divide by 9) → y - x = 3 ...(2)

Step 2: Add Equations

(x - x) + (y + y) = 9 + 3
2y = 12 → y = 6

Step 3: Find x

Put y = 6 in (1): x + 6 = 9 → x = 3

Number = 36

Similar to Q2(iv)

A person withdrew ₹4000. He got ₹100 and ₹500 notes only. The total number of notes is 20. Find the number of notes of each type.

View Step-by-Step Solution

Step 1: Form Equations

Let ₹100 notes = x, ₹500 notes = y.
Total notes: x + y = 20 ...(1)
Total Value: 100x + 500y = 4000 (Divide by 100) → x + 5y = 40 ...(2)

Step 2: Eliminate x

Subtract (1) from (2):
(5y - y) = 40 - 20
4y = 20 → y = 5

Step 3: Find x

Put y = 5 in (1): x + 5 = 20 → x = 15

₹100 notes = 15, ₹500 notes = 5

Similar to Q2(v)

A library has a fixed charge for the first 2 days and an additional charge for each day thereafter. Person A paid ₹22 for a book kept for 6 days, while Person B paid ₹14 for a book kept for 4 days. Find the fixed charge and charge per extra day.

View Step-by-Step Solution

Step 1: Form Equations

Let fixed charge (2 days) = x, per day charge = y.
Person A (6 days = 2 fixed + 4 extra): x + 4y = 22 ...(1)
Person B (4 days = 2 fixed + 2 extra): x + 2y = 14 ...(2)

Step 2: Eliminate x

Subtract (2) from (1):
2y = 8 → y = 4

Step 3: Find x

Put y = 4 in (2): x + 8 = 14 → x = 6

Fixed Charge = ₹6, Charge per extra day = ₹4

Q1. Solve the Equations

Q2. Word Problems