Hard Level Worksheet Questions

Part A: Subjective Questions

(1) If 3x – 7 = 2x + 5, find x.

Solving x =

Awesome! Subtract 2x: 3x - 2x = 5 + 7, so x = 12.

(2) Solve: x+32 = x−23

Solving: x =

Great job! Cross multiply: 3(x + 3) = 2(x - 2), so x = 13.

(3) What should be added to both sides of the equation: 2x – 1 = 3x – 5 to make it solvable easily?

Perfect! Adding 4 gives: 2x + 3 = 3x - 1.

(4) The difference between two numbers is 7. If 4 times the smaller number is equal to 3 times the larger number, find the numbers.

Let smaller number = x, larger =

Smaller = , Larger =

Excellent! The numbers are 21 and 28.

(5) Solve for x: 2x−13 + x+22 = 7

Solving: x =

(6) If x – 1 = 2(x + 3) – 5, find x.

Solving: x =

That's correct! Expand: x - 1 = 2x + 1, so x = 0.

(7) Solve: 3(2x – 1) – 4(x + 2) = 5

Solving: x =

Well done! Expand: 6x - 3 - 4x - 8 = 5, so x = 8.

(8) If x−34 + 2x+13 = 3, find x.

Solving: x =

Brilliant! Multiply by 12 to clear fractions.

(9) Find x: 5x−26 = x+13

Solving: x =

You nailed it! Cross multiply: 5x - 2 = 2(x + 1), so x = 2.

(10) What is the solution of: x−45 + 3x+210 = 4

Solving: x =

Perfect! Multiply by 10 to clear fractions, then solve.

Drag each equation to its most appropriate solution method:

(1) Solve: 3x−52 – x+13 = x−24

Solving: x =

Excellent! x = 2811 or approximately 2.55.

(2) The denominator of a fraction is 3 more than the numerator. If 5 is added to both numerator and denominator, the fraction becomes 3/4. Find the original fraction.

Let numerator = x

Then denominator =

Original fraction = xx+3

After adding 5: ()/(x+8x+3) =

Solving: x =

Original fraction =

Perfect! The original fraction is 47.

(3) A man is 4 times as old as his son. After 5 years, he will be 3 times as old as his son. Find their present ages.

Let son's present age = x years

Father's present age = years

After 5 years:

Son's age = years

Father's age = years

Given condition: 4x + 5 = (x + 5)

Solving: x =

Therefore, Father = years, Son = years

Great work! Father is 40 years and son is 10 years old.

(4) A shopkeeper sells an item at 20% profit. If the cost price is ₹x and selling price is ₹(x + 80), find the cost price.

Cost price = ₹x

Selling price = ₹

Profit = Selling priceCost price - Cost priceSelling price =

Profit% = (Profit/Cost priceSelling price) × 100 = %

So: (/) × 100 = 20

Solving: Cost price =

Excellent! The cost price is ₹400.

(5) A number when multiplied by 4 and reduced by 3 gives the same result as when multiplied by 2 and increased by 5. Find the number.

Let the number = x

First condition: 4x -

Second condition: 2x +

Since both results are same: 4x - 3 =+- 2x + 5

Solving: x =

Perfect! The number is 4.

(1) Solve the following equation step-by-step and check your solution: x+12 + 2x−34 = 3x+56

To solve, find of denominators.

By solving: x =

Excellent! x = 136 is the solution.

(2) The difference between the ages of a father and son is 28 years. If 4 years ago, the father's age was 4 times the son's age, find their present ages.

Let son's present age = x years

Father's present age = years

4 years ago:

Son's age = years

Father's age = years

Given condition: x+24 = ×

Solving: x = (Round to nearest integers)

Therefore, Son’s age is years and Father’s age is years

Perfect! Father is 41 years and son is 13 years old.

(3) A piece of rod is broken into two parts such that the longer part is 6 cm more than the shorter part. If one-fifth of the longer part is equal to one-third of the shorter part, find the length of the whole rod.

Let shorter part = x cm

Longer part = cm

Given condition: (x + 6) = (x)

Solving: x =

Therefore, Shorter part is cm and Longer part = cm

Where Total length = 9 + 15 = cm

Outstanding! The whole rod is 24 cm long.

(4) A number when divided by 2 gives a result which is 2 less than when the same number is divided by 3. Find the number.

Let the number = x

When divided by 2:

When divided by 3:

Given condition: x2 = x3 -

Solving: x =

Since the problem might expect positive answer: x =

Fantastic! The number is 12.

Part B: Objective Questions

Test your understanding with these challenging multiple choice questions:

1. If 5x – 3 = 2x + 12, then x =

(a) 5 (b) 3 (c) 4 (d) 6

Super job! 5x - 2x = 12 + 3, so x = 5.

2. In a linear equation ax + b = 0, the solution is given by:

(a) x = a – b (b) x = –b/a (c) x = −ab (d) x = ba

Well done! ax = -b, so x = −ba.

3. Which of these equations has solution x = –1?

(a) 2x + 3 = 1 (b) x – 1 = 2 (c) x + 1 = 0 (d) 3x = –3

That's right! x + 1 = 0 gives x = -1.

4. A rational number is such that when 2 is added to both numerator and denominator, it becomes 3/4. What is the original number?

(a) 12 (b) 35 (c) 23 (d) 25

Correct! 1+22+2 = 34, so original is 12.

5. A number is subtracted from twice itself, and the result is 18. The number is:

(a) 6 (b) 9 (c) 12 (d) 18

Fantastic! 2x - x = 18, so x = 18.

🎉 Remarkable Achievement! You've Mastered Hard-Level Linear Equations!

Here's what you conquered:

• Complex multi-step equations with fractions and brackets
• Advanced cross-multiplication techniques for fractional equations
• Sophisticated word problems involving age, money, and geometric relationships
• Multi-variable relationship problems (consecutive numbers, ratios)
• Advanced algebraic manipulation and solution verification
• Understanding general solution formulas for linear equations
• Complex real-world applications requiring equation setup and solving

Your advanced linear equation skills prepare you for algebra, systems of equations, and advanced mathematical problem-solving!