13.7 Going Deeper: Enrichment & Exam Preparation

Where Algebra Plays in Real Life

Math in the Real World

  • Budgeting and break-even. A chai stall pays a fixed rent each day and a small cost per cup. Writing “profit = (\text{price} - \text{cost per cup})\times n - \text{rent}” turns a money question into a linear equation you can solve for the cups n needed to break even, exactly the kind of formula a shopkeeper or start-up uses every day.
  • Spreadsheets and coding. A spreadsheet cell such as =2*A1+3 is the expression 2x+3 with x standing for cell A1. Every formula in a budget sheet, and every line like total = 2*price + 3 in a program, is algebra wearing a different costume.
  • Science formulas. Speed = \dfrac{\text{distance}}{\text{time}}, or the cost of a taxi = 50 + 12k for k kilometres, are linear relationships. Rearranging them to find an unknown is “solving a linear equation.”
  • Puzzles and games. “Think of a number” tricks, Sudoku-style logic, and the magic of guessing a friend’s birthday all run on algebra, a letter-number quietly cancelling or being recovered from a final total.
  • A bridge to later chapters. Forming and solving linear equations here prepares you for graphing straight lines and for simultaneous equations, where two unknowns are pinned down at once.

Quick Reference: Forming and Solving Linear Equations

Knowing the standard steps by heart turns a wordy problem into a quick, confident solution.

Step What to do Example: “5 more than thrice a number is 26”
1 Name the unknown with a letter let the number be x
2 Translate each phrase into algebra thrice it =3x; 5 more =3x+5
3 Write the equation 3x + 5 = 26
4 Undo the addition (subtract) 3x = 21
5 Undo the multiplication (divide) x = 7
6 Check in the original words 3(7)+5 = 26 ✓

Many word problems hinge on translating English phrases into expressions. Keep this table handy:

In words In algebra
a number x
3 more than twice a number 2x + 3
5 less than a number x - 5
the number, doubled, then 7 added 2x + 7
half of a number, less 4 \dfrac{x}{2} - 4
a two-digit number, tens t, units u 10t + u
consecutive integers x,\ x+1,\ x+2
age “in 5 years” x + 5

Use the solver below to practise solving any equation of the form ax + b = c:

Linear equation solver: ax + b = c

Exam Tip When you “do the same thing to both sides,” undo operations in reverse order: peel off addition/subtraction first, then multiplication/division. To solve 3x + 5 = 26, subtract 5 before dividing by 3. Reversing that order is a classic slip that costs marks. And always finish by substituting your answer back into the original equation, a 10-second check that catches almost every arithmetic error.

Memory Tricks & One-Page Revision

Quick Revision Card

  • Why a number trick works: write each step in x; the trick is foolproof exactly when x cancels from the final expression.
  • Date trick: the recipe lands on 100M + 165 + D. Subtract 165; the last two digits are the day D, the rest is the month M.
  • Pyramid apex from base a,b,c is a + 2b + c; from a,b,c,d it is a + 3b + 3c + d (weights 1,2,1 and 1,3,3,1, Pascal’s triangle).
  • 2\times 2 calendar block sums to 4a + 16, where a is the top-left number, so a = \dfrac{\text{sum} - 16}{4}.
  • Solving ax + b = c: first ax = c - b, then x = \dfrac{c-b}{a}. Undo +/- before \times/\div.
  • Place-value tricks: swap-and-subtract a 2-digit number gives 9(b-a); abc + bca + cab = 111(a+b+c); abcabc = abc\times 7\times 11\times 13.
  • Word-problem drill: name the unknown, translate phrase-by-phrase, form the equation, solve, then check in words.

Spot the Mistake

Common Exam Mistakes

  • Dividing before subtracting: from 3x + 5 = 26, writing “x + 5 = \tfrac{26}{3}” is wrong, subtract the 5 first, getting 3x = 21.
  • Mistranslating “3 more than twice a number” as 3 + 2 + x or 2(x+3). It is 2x + 3.
  • In age problems, forgetting to age both people: “in 5 years” turns x into x+5 and 7x into 7x+5, not just one of them.
  • Writing a two-digit number with tens t and units u as “tu” or “t + u” instead of the correct place value 10t + u.
  • Solving for the letter but forgetting the question, e.g. finding the son’s age when the problem asked for the father’s.

Equation Duel

A quick drill: a linear equation appears, you type the value of x, and you build a streak.

Solve-it streak
Solve for x, type the answer, press Check.
Score 0 · Streak 0

Exam Corner: CBSE-Style Practice

Mixed Practice (objective, short, long, HOTS)

Objective type (1 mark each)

  1. The phrase “3 more than twice a number x” is written as ______.
  2. Solve: 4x = 28, so x = ______.
  3. Solve: \dfrac{x}{5} = 6, so x = ______.

Short answer (2 marks each)

  1. Solve the equation 7x - 9 = 4x + 12.
  2. Run this trick in algebra: think of a number, triple it, add 9, divide by 3, then subtract the original number. Show that the result is always the same, and state it.

Long answer (3 marks each)

  1. A father is 5 times as old as his son. In 6 years he will be only 3 times as old. Find their present ages.
  2. A two-digit number has digit-sum 11. When its digits are reversed, the new number is 27 more than the original. Find the number.

HOTS (Higher Order Thinking)

  1. In a three-row number pyramid each box is the sum of the two below it. The apex is 100, the left base number is 20 and the middle base number is 30. Find the right base number.
  2. A “think of a number” trick says: choose a number, add 4, double the result, subtract 6, halve it, then subtract the number you first thought of. Prove with algebra that the answer is always the same, and find it.

Assertion–Reason (Choose: (a) both true, R explains A; (b) both true, R does not explain A; (c) A true, R false; (d) A false, R true.)

  1. Assertion (A): The solution of 2x + 5 = 17 is x = 6.   Reason (R): To solve a linear equation we undo addition/subtraction first, then multiplication/division.
  1. \mathbf{2x + 3}, “twice the number” is 2x, and “3 more” adds 3.
  2. 4x = 28 \Rightarrow x = 28 \div 4 = \mathbf{7}.
  3. \dfrac{x}{5} = 6 \Rightarrow x = 6 \times 5 = \mathbf{30}.
  4. 7x - 9 = 4x + 12 \Rightarrow 7x - 4x = 12 + 9 \Rightarrow 3x = 21 \Rightarrow x = \mathbf{7}. (Check: 7(7)-9 = 40 and 4(7)+12 = 40 ✓.)
  5. Let the number be x: x \xrightarrow{\times 3} 3x \xrightarrow{+9} 3x+9 \xrightarrow{\div 3} x+3 \xrightarrow{-x} \mathbf{3}. The x cancels, so the answer is always 3.
  6. Let the son be s; the father is 5s. In 6 years: 5s + 6 = 3(s + 6) = 3s + 18 \Rightarrow 2s = 12 \Rightarrow s = 6. Son 6 years, father 30 years. (Check: in 6 years 36 = 3\times 12 ✓.)
  7. Let the tens digit be t and units be u, so the number is 10t + u. Given t + u = 11 and (10u + t) - (10t + u) = 27 \Rightarrow 9u - 9t = 27 \Rightarrow u - t = 3. Solving t + u = 11 and u - t = 3 gives u = 7,\ t = 4. The number is \mathbf{47} (check: 74 - 47 = 27, 4 + 7 = 11 ✓).
  8. With base a, b, c the apex is a + 2b + c = 100. Here a = 20 and b = 30, so 20 + 2(30) + c = 100 \Rightarrow 80 + c = 100 \Rightarrow c = \mathbf{20}.
  9. Let the number be x: x \xrightarrow{+4} x+4 \xrightarrow{\times 2} 2x+8 \xrightarrow{-6} 2x+2 \xrightarrow{\div 2} x+1 \xrightarrow{-x} \mathbf{1}. The x cancels, so the answer is always 1.
  10. (a), Both are true and R explains A: 2x + 5 = 17 \Rightarrow 2x = 12 \Rightarrow x = 6, found by undoing the “+5” first and then the “\times 2.”

Connections to Other Chapters

How This Chapter Links Forward

  • Linear equations in one variable: the forming-and-solving skill here is the heart of the dedicated equations chapter, where the same steps tackle equations with the unknown on both sides and with brackets.
  • Graphs and straight lines: an expression like 2x + 3 becomes a line when plotted, and “solving an equation” turns into “reading off where two lines meet.”
  • Mensuration and science: perimeter, area and speed–distance–time formulas are all expressions waiting to be rearranged, the very algebra practised in this chapter.

Glossary

Key Terms

  • Variable (letter-number): a symbol such as x that stands for “any number you choose.”
  • Expression: a combination of numbers, variables and operations, e.g. 2x + 3, no equals sign.
  • Equation: a statement that two expressions are equal, e.g. 3x + 5 = 26, which can be solved for the unknown.
  • Linear equation: an equation in which the variable appears only to the first power, like ax + b = c.
  • Coefficient: the number multiplying a variable; in 3x the coefficient is 3.
  • Constant: a fixed number with no variable, like the 5 in 3x + 5.
  • Solution (root): the value of the variable that makes the equation true.
  • Place value form: writing a two-digit number as 10t + u, the key to most digit tricks.