Linear Equations – Complete Notes for Competitive Exams

1. Introduction

Linear equations are algebraic equations of the first degree. They involve variables raised only to the power 1.
These are very common in Quantitative Aptitude.

Standard Form

$$ax + b = 0$$ax+b=0

where $a \neq 0$a=0

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2. Types of Linear Equations

1. Single-variable linear equation
   • Example: $2x + 5 = 13$2x+5=13
   • Solve for x
2. Two-variable linear equations
   • Example: $2x + 3y = 12$2x+3y=12 and $x – y = 2$x−y=2
   • Solved using substitution or elimination
3. Three-variable linear equations
   • Example: $x + y + z = 6$x+y+z=6, $2x – y + z = 3$2x−y+z=3, $x + 2y – z = 4$x+2y−z=4
   • Solve using substitution or Cramer’s Rule

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3. Basic Formulas / Rules

For single variable:

$$ax + b = 0 \implies x = -\frac{b}{a}$$ax+b=0⟹x=−ab​

For two variables:

• Substitution Method
  Solve one equation for one variable, then substitute in the other
• Elimination Method
  Add/subtract equations to eliminate a variable
• Cross-Multiplication Formula (for $a_1x + b_1y + c_1 = 0$a1​x+b1​y+c1​=0 and $a_2x + b_2y + c_2 = 0$a2​x+b2​y+c2​=0)

$$\frac{x}{b_1c_2 – b_2c_1} = \frac{y}{c_1a_2 – c_2a_1} = \frac{1}{a_1b_2 – a_2b_1}$$b1​c2​−b2​c1​x​=c1​a2​−c2​a1​y​=a1​b2​−a2​b1​1​

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4. Solving Tips

• Check for unique, infinite, or no solution
• For substitution/elimination, eliminate carefully
• For word problems, convert statements → equations → solve
• Always verify by substituting values

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5. Common Exam Applications

• Age problems
• Ratio problems
• Profit and loss
• Mixtures
• Speed, distance, time
• Number problems

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Top 25 Practice Questions – Linear Equations

Q1.

Solve: $2x + 5 = 13$2x+5=13

Q2.

Solve: $5x – 7 = 18$5x−7=18

Q3.

Solve: $3x + 4 = 19$3x+4=19

Q4.

Solve: $2x + 3y = 12, x – y = 2$2x+3y=12,x−y=2

Q5.

Solve: $x + y = 10, x – y = 4$x+y=10,x−y=4

Q6.

Solve: $2x + 3y = 16, 3x – y = 5$2x+3y=16,3x−y=5

Q7.

Solve: $x + y + z = 6, 2x – y + z = 3, x + 2y – z = 4$x+y+z=6,2x−y+z=3,x+2y−z=4

Q8.

The sum of two numbers is 15 and their difference is 3. Find the numbers.

Q9.

A shopkeeper sold 5 pencils and 3 pens for ₹50. 2 pencils and 4 pens cost ₹26. Find the cost of each pencil and pen.

Q10.

Solve: $4x – 5 = 11$4x−5=11

Q11.

Solve: $7x + 3 = 31$7x+3=31

Q12.

Solve: $2(x + 3) = 14$2(x+3)=14

Q13.

Solve: $5x – 2(x – 3) = 16$5x−2(x−3)=16

Q14.

Solve: $3x + 2y = 12, 5x – y = 13$3x+2y=12,5x−y=13

Q15.

Solve: $x + 2y = 8, 2x – y = 3$x+2y=8,2x−y=3

Q16.

The sum of three numbers is 60. First number = second number + 4. Third number = twice the second. Find the numbers.

Q17.

Solve: $x/2 + y/3 = 5, x – y = 4$x/2+y/3=5,x−y=4

Q18.

Solve: $3x + 5 = 20$3x+5=20

Q19.

Solve: $4x – 3 = 21$4x−3=21

Q20.

Two numbers are in the ratio 3:5. Sum = 64. Find the numbers.

Q21.

Solve: $x + y + z = 12, x + y – z = 4, x – y + z = 6$x+y+z=12,x+y−z=4,x−y+z=6

Q22.

Solve: $6x – 3y = 9, 2x + y = 8$6x−3y=9,2x+y=8

Q23.

Solve: $2x + 5y = 20, 3x – y = 4$2x+5y=20,3x−y=4

Q24.

The sum of ages of A and B is 30. Five years ago, sum = 20. Find current ages.

Q25.

Solve: $x + y = 18, x – y = 4$x+y=18,x−y=4

Ans

Answers – Linear Equations

Q1. $x = 4$x=4
Q2. $x = 5$x=5
Q3. $x = 5$x=5
Q4. $x = 3, y = 3$x=3,y=3
Q5. $x = 7, y = 3$x=7,y=3
Q6. $x = 2, y = 4$x=2,y=4
Q7. $x = 2, y = 1, z = 3$x=2,y=1,z=3
Q8. Numbers: 9 and 6
Q9. Pencil = ₹4, Pen = ₹6
Q10. $x = 4$x=4
Q11. $x = 4$x=4
Q12. $x = 4$x=4
Q13. $x = 5$x=5
Q14. $x = 3, y = 1.5$x=3,y=1.5
Q15. $x = 4, y = 2$x=4,y=2
Q16. Numbers: 16, 12, 24
Q17. $x = 8, y = 4$x=8,y=4
Q18. $x = 5$x=5
Q19. $x = 6$x=6
Q20. Numbers: 24 and 40
Q21. $x = 4, y = 2, z = 6$x=4,y=2,z=6
Q22. $x = 2, y = 4$x=2,y=4
Q23. $x = 4, y = 2.4$x=4,y=2.4
Q24. Ages: A = 17, B = 13
Q25. $x = 11, y = 7$x=11,y=7