Skip to lesson content

Lesson 13 of 13

Finding the Unknown · Lesson 13 of 13

Chapter Summary and Practice

“Connect the chapter’s ideas and use them confidently in mixed problems.”

Learning Objectives

• Connect balances, equations, patterns, machines, and word problems. • Choose justified operations and check original equalities. • Distinguish numerical solutions from contextually possible answers. • Recognise common algebra errors and explain a number trick. • Apply the chapter’s ideas in substantial mixed practice.

One idea connects the chapter

The chapter begins with equal weights and ends with mathematical relationships written using unknown letters. The central idea remains equality. A balanced scale says two weights match; an equation says two expression values match. A solution makes that statement true. Solving reveals the unknown by making equal, reversible changes to both sides.

Trial and error checks candidate values, while systematic solving chooses operations that isolate the unknown. Neither method changes the definition of a solution. An answer must pass substitution into the original equation. A story adds further conditions: a position or a count must be a suitable whole number, while a measured length may be fractional.

Chapter ideaRelationship to rememberWhat to check
Hanging scalesA balanced total is shared equally; nested branches balance locallyDo not confuse the whole total with one branch
Equation languageUnknown; LHS; RHS; solution by substitutionAn expression alone is not an equation
Trial and inverse operationsTrials compare values; inverse operations undo changesWhole-number trials can miss fractions
Equality operationsAdd/subtract equally; multiply/divide by a non-zero number for reversible solvingApply the operation to entire sides
Unknowns on both sidesSubtract matching unknown groups; combine like termsNever divide blindly by an unknown
Zero and no solution2s = 0 gives s = 0; 4 = 5 is a contradictionZero must still satisfy the original
PatternsTriangle sticks 2n + 1; T tiles 3k + 1; cap sequence 3n + 3; staircase sticks 9n + 4Count shared edges and use whole positions
Everyday equationsInitial amount + repeated additions; fixed fee + unit chargeDefine what the letter counts and its unit
Tricks and chainsCarry a letter through each operation; reverse order to undoBrackets preserve the intended grouping
Mistake correctionFind the first invalid step, repair, substituteSome unfamiliar chains are valid
History and general ruleNotation changes; x = (D − B)/(A − C) if A ≠ CInclude signed constants; denominator cannot be zero
Diagrams and machinesSerial routes differ from parallel routes; lengths and angles supply totalsCount all pieces and use given equalities
Connected challengesExpress related quantities using one unknown; transform given equalities directlyCheck every relationship, not only the total

Choose a strategy that fits the relationship

Begin by identifying the unknown and the relationship you are given. A simple grouped equation may be quickest by undoing outer operations, while an equation with the unknown in both brackets may be clearer after expansion. A related-expression question may not need the unknown at all. State why your choice works and keep a check at the end.

Return to the balance model

Problem
Five equal sacks balance two sacks plus 21 kg. Find one sack using an equation.

  1. 1.Let s be one sack’s weight in kilograms. Write 5s = 2s + 21.
  2. 2.Subtract 2s: 3s = 21. Divide by 3: s = 7 kg.
  3. 3.Check: 5 × 7 = 35 kg and 2 × 7 + 21 = 35 kg. This is equal removal followed by sharing equally.
Choose an efficient grouped method

Problem
Solve 6(x + 2) + 12 = 54.

  1. 1.Subtract 12: 6(x + 2) = 42. Divide by 6: x + 2 = 7.
  2. 2.Subtract 2: x = 5.
  3. 3.Check: 6(5 + 2) + 12 = 42 + 12 = 54. Expanding would also work, but undoing is short here.
An unknown on both sides with brackets

Problem
Solve 3(a + 4) = 2a + 7.

  1. 1.Expand the left: 3a + 12 = 2a + 7.
  2. 2.Subtract 2a: a + 12 = 7. Subtract 12: a = −5.
  3. 3.Check: left = 3(−5 + 4) = −3; right = 2(−5) + 7 = −3. A negative solution is allowed.
An equation answer and a pattern answer

Problem
A T shape has rule T = 3k + 1. Could it have 41 tiles?

  1. 1.Write 3k + 1 = 41. Subtract 1: 3k = 40. Divide by 3: k = 40/3.
  2. 2.This is a numerical solution of the equation because 3 × (40/3) + 1 = 41.
  3. 3.But it is not a whole position. Hence there is no complete T shape with 41 tiles in this sequence.
Combine a relationship and a total

Problem
Two people save ₹900 together. One has ₹150 more than the other. Find their savings.

  1. 1.Let the smaller savings be r rupees. The larger is r + 150.
  2. 2.Write r + (r + 150) = 900. Combine: 2r + 150 = 900.
  3. 3.Subtract 150: 2r = 750. Divide by 2: r = ₹375. The larger savings are ₹525.
  4. 4.Check: 375 + 525 = 900 and 525 − 375 = 150.
Recover a parallel-machine input

Problem
A machine outputs 3x − (x + 3). What input gives 17?

  1. 1.Write 3x − (x + 3) = 17. Expand the subtraction: 2x − 3 = 17.
  2. 2.Add 3: 2x = 20. Divide by 2: x = 10.
  3. 3.Check the branches: 3 × 10 = 30; 10 + 3 = 13; 30 − 13 = 17.

Explain the magic, rather than only trying it

Try this sequence: think of a number, multiply by 2, add 10, divide by 2, subtract your original number, and add 3. The result is always 8. Checking a few starting numbers gives evidence, but carrying an unknown through the process explains why every starting number works. The original number disappears by subtraction.

Why the result is always eight

Problem
Show algebraically that the closing number trick always gives 8.

  1. 1.Let the original number be x. Multiplying by 2 gives 2x; adding 10 gives 2x + 10.
  2. 2.Dividing the entire result by 2 gives (2x + 10)/2 = x + 5.
  3. 3.Subtracting the original number gives (x + 5) − x = 5. Adding 3 gives 8.
  4. 4.For x = −4, the values are −4, −8, 2, 1, 5, 8. For x = 1/2, they are 1/2, 1, 11, 11/2, 5, 8. The cancellation works for both.

To create your own fixed-result trick, arrange for the final expression to contain +x and −x so that they cancel. For example, multiply a starting number by 3, add 12, divide by 3, subtract the original number, then add 2. The expression becomes x + 4 − x + 2 = 6. This uses the same grouping and inverse-operation ideas as equation solving.

Check for the errors that matter

A reliable final review asks whether every operation affected both sides, every bracket term was included, and every sign retained its meaning. Then substitute the proposed answer and check the situation. A zero answer is valid if it balances; a contradiction is not an answer. These habits are more reliable than expecting all solutions to be neat positive whole numbers.

Common mistake

Do not divide only an unknown term while leaving a constant unchanged. Do not treat 4w as 4, or assume multiplying a negative bracket keeps its inner signs. In the general rule, keep signed constants and require a non-zero denominator. Reject fractional counts only when the situation requires whole items or positions.

Quiz

Quick check

Which statement best defines a solution?

Quick check

For x + 7 = x + 9, what follows after subtracting x?

Quick check

Which transformation correctly divides 6x + 12 = 30 by 6?

Quick check

Which stick rule describes the staircase sequence B?

Quick check

What do matching side strokes on an isosceles triangle allow you to conclude?

Quick check

Which equation models a ₹90 fixed fee plus ₹15 per item, costing ₹300?

Quick check

What does dividing by an unknown in 5s = 3s risk losing?

Quick check

What does the closing trick do to the original x?

Practice Problems

Practice Problems
  1. Draw a balanced hanging scale of total weight 30. One branch has two known 4-unit objects and an unknown. Find the unknown and explain the total-versus-branch distinction.
  2. Use a trial table and then inverse operations to solve 2n + 1 = 63. Compare the two methods.
  3. Solve and check 7x − 5 = 23, y/12 = −3, and −4z = 9.
  4. Solve 5(a + 2) = 3a − 6 and check both original sides.
  5. Decide whether 8s = 5s and t + 6 = t + 7 have a zero solution or no solution. Explain.
  6. Find the position of a 121-tile T. Show both a three-arm count and a row-and-stem count.
  7. For sequence A, find the stick count at position 15. For sequence B, determine whether 112 sticks is possible.
  8. A meal costs ₹40 per person plus ₹80 delivery. There are four family members and a ₹640 budget. Find how many friends can join; use both total-people and friends-only unknowns.
  9. Two savings totals are 2000 + 300m and 3200 + 200m. Find when they meet and their common total.
  10. Create four different equations with solution u = 6. Explain one forward and one reverse equation chain.
  11. Find the first error in 4x + 8 = 20 → x + 8 = 5. Repair the chain and check.
  12. Derive the general rule for Ax + B = Cx + D, state its restriction, and use it on 8x − 3 = 5x + 12.
  13. A serial machine doubles an input then adds 6. A parallel machine subtracts the entire quantity (input + 3) from three times the input. Write each rule and find the input that gives 21.
  14. A triangle’s angles are b°, (b + 15)°, and (b − 15)°. Find all three.
  15. Given 10p + 6 = 46, find 5p + 2 by transforming the equality directly.
  16. Children and donkeys have 32 heads and 92 feet. Find both counts and check.
  17. Explain the always-eight trick with a letter, then create and justify your own trick with a different fixed result.
  18. Explain how an older symbol for an unknown and a modern letter can describe the same relationship.

Key Takeaways

Key Takeaways

• Equality connects balances, equations, patterns, stories, and machines. • Define the unknown and keep its units and meaning clear. • Choose reversible equal operations and show the reasoning. • Check substitution, signs, brackets, and contextual restrictions. • Use letters to explain a whole pattern or trick, not just one numerical example.

Previous · Lesson 12

Connected Word Problems and Challenges

Next

End of chapter