12  Linear Equations

12.1 Introduction

Linear equations form the backbone of algebra and quantitative aptitude.
They represent relationships between variables with degree one (highest power of variable = 1).
In exams, linear equations appear in forms of simple solving, word problems, equations in two or more variables, and applications in ages, mixtures, percentages, and time–work–distance.

12.2 1) Basics of Linear Equations

  • General form (one variable):
    \[ax+b=0\] where \(a\neq0\).
    Solution: \(x=-\frac{b}{a}\).

  • General form (two variables):
    \[ax+by+c=0\]
    Represents a straight line in the coordinate plane.

  • Linear equation in \(n\) variables:
    \[a_1x_1+a_2x_2+\dots+a_nx_n+b=0\]

12.3 2) Types of Solutions (Two Variables)

System of two equations:
\[a_1x+b_1y=c_1\]
\[a_2x+b_2y=c_2\]

  • Unique solution (intersecting lines):
    \(\frac{a_1}{a_2}\neq \frac{b_1}{b_2}\)

  • No solution (parallel lines):
    \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}\)

  • Infinite solutions (coincident lines):
    \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\)

12.4 3) Methods of Solving

12.4.1 3.1 Substitution Method

  • Express one variable in terms of another and substitute.
  • Example: Solve \(2x+y=10\), \(x-y=1\).
    From second: \(x=y+1\). Sub into first: \(2(y+1)+y=10 \Rightarrow 3y+2=10 \Rightarrow y=8/3\), \(x=11/3\).

12.4.2 3.2 Elimination Method

  • Multiply equations if needed and eliminate one variable.
  • Example: Solve \(3x+2y=16\), \(2x+3y=13\).
    Multiply first by 3: \(9x+6y=48\).
    Multiply second by 2: \(4x+6y=26\).
    Subtract: \(5x=22 \Rightarrow x=22/5\), then \(y=(16-3x)/2=14/10=7/5\).

12.4.3 3.3 Cross Multiplication (for two variables)

For system \(a_1x+b_1y+c_1=0\), \(a_2x+b_2y+c_2=0\):

\[ \frac{x}{b_1c_2-b_2c_1} = \frac{y}{c_1a_2-c_2a_1} = \frac{1}{a_1b_2-a_2b_1} \]

12.5 4) Applications in Word Problems

12.5.1 4.1 Age Problems

Father is 3 times son’s age. In 10 years, difference = 20. Find ages.
Let son=x, father=3x. After 10 yrs: (3x+10)−(x+10)=2x=20 ⇒ x=10, father=30.

12.5.2 4.2 Mixture Problems

A fruit seller mixes 10 kg apples at 80/kg with 15 kg apples at 100/kg. Selling price = 96/kg. Find profit/loss.
CP = (10×80+15×100)/(25)=92. SP=96 ⇒ Profit=4 on 92=4.35%.

12.5.3 4.3 Time–Work

A does work in 6 days, B in 8 days. Together in 1 day?
Rate: 1/6+1/8=7/24 ⇒ total time=24/7≈3.43 days.

12.5.4 4.4 Percent/Ratio Problems

Twice a number plus thrice another = 36. Their difference=6. Find numbers.
Let x,y. 2x+3y=36, x−y=6 ⇒ Solve.

12.6 5) Linear Equations in Three Variables

System:
\[a_1x+b_1y+c_1z=d_1\]
\[a_2x+b_2y+c_2z=d_2\]
\[a_3x+b_3y+c_3z=d_3\]

  • Solve using elimination/substitution or determinants (Cramer’s rule).
  • Exam questions often simple, solvable by stepwise substitution.

12.7 6) Special Forms

  • Equations reducible to linear: e.g., \(\frac{1}{x}+\frac{1}{y}=3\), \(\frac{1}{x}-\frac{1}{y}=1\) → substitute \(p=1/x\), \(q=1/y\).
  • Word equations: “If you double the number and add 5, result is 21” → \(2x+5=21\).

12.8 7) Solved Examples

Ex 1 Solve: \(x+2y=8\), \(3x-y=5\).
From first: \(x=8-2y\). Sub: \(3(8-2y)-y=5 \Rightarrow 24-6y-y=5 \Rightarrow y=19/7\), \(x=30/7\).

Ex 2 A and B together earn 1,200. A earns 200 more than B. Find amounts.
Let B=x ⇒ A=x+200. Then x+x+200=1200 ⇒ 2x=1000 ⇒ x=500, A=700.

Ex 3 Solve by cross multiplication: 2x+3y=8, 4x−y=2.
Cross method ⇒ x=1, y=2.

Ex 4 In 3 variables: x+y+z=6, x+2y+3z=14, 2x+y+z=10.
From first: x=6−y−z. Sub into others → solve → (x,y,z)=(3,1,2).

12.9 8) Practice Set – Level 1

  1. Solve: 5x+2y=20, 3x−y=1.
  2. A number plus its double equals 21. Find number.
  3. Two numbers differ by 4. Their sum=20. Find numbers.
  4. A pen and pencil cost 18, two pens and three pencils cost 48. Find cost of each.
  5. Solve: x+y=10, x−y=4.

12.10 Practice Set – Level 2

  1. 3x+4y=25, 5x+6y=41. Solve.
  2. A father is twice his son’s age. In 10 yrs sum of ages=80. Find ages.
  3. Three numbers add to 60. Their ratio 2:3:5. Find numbers.
  4. Solve: 2x+y+z=8, x−y+2z=3, 3x+2y−z=7.
  5. The cost of 2 apples and 3 bananas=40, 4 apples and 9 bananas=100. Find apple and banana cost.

12.11 9) Common Mistakes

  • Forgetting to check solution type (unique/no/infinite).
  • Mixing CP, SP context with linear algebra.
  • Arithmetic slips in substitution/elimination.
  • Misreading word problems (e.g., “more than,” “less than”).

12.12 Summary

  • Linear equations = degree 1 relations.
  • Solving methods: substitution, elimination, cross multiplication.
  • Word problems → translate carefully into equations.
  • In 3 variables, use substitution/determinants.
  • Key exam skill: speed in setting up + eliminating without mistakes.