Linear Equations and Linear Inequalities
Linear Equations and Linear Inequalities
A linear equation is a statement of balance: whatever you do to one side you do to the other. NTPC turns word statements (marks, coins, tickets) into one equation, asks systems of two or three equations, tests when a system has no solution, and checks inequalities, where a negative multiplier reverses the sign.
1. Rules Box
One variable: ax + b = c gives x = (c − b)/a
Two variables a₁x + b₁y = c₁ and a₂x + b₂y = c₂: eliminate one variable by matching coefficients, or substitute
| Condition on the two lines | Meaning | Number of solutions |
|---|---|---|
| a₁/a₂ ≠ b₁/b₂ | Lines cross | Exactly one (unique) |
| a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | Parallel lines | None (inconsistent) |
| a₁/a₂ = b₁/b₂ = c₁/c₂ | Same line | Infinitely many |
2. Shortcut Rules
| Situation | Rule |
|---|---|
| Three equations, each missing one variable or with ±1 coefficients | Add or subtract pairs to remove a variable fast |
| Equations in 1/x and 1/y | Put u = 1/x, v = 1/y; solve the linear system in u, v |
| All attempted, +p for right, −q for wrong, n questions, score S | Correct = (S + qn)/(p + q) |
| Inequality multiplied or divided by a negative number | Reverse the sign |
| Option checking | Substitute the options into both equations; it is often faster |
✗ −2x > 6, so x > −3 | ✓ Dividing by −2 reverses the sign: x < −3
✗ kx + 3y = 6 and 4x + 6y = 9 have no solution when k/4 = 6/9 | ✓ No solution needs a₁/a₂ = b₁/b₂: k/4 = 3/6, so k = 2
हिंदी नोट: रैखिक समीकरण में दोनों पक्षों पर एक जैसी क्रिया कीजिए। असमिका को ऋणात्मक संख्या से गुणा या भाग करने पर चिह्न उलट जाता है। a₁/a₂ = b₁/b₂ ≠ c₁/c₂ हो तो कोई हल नहीं होता।
Exam Pointer: Verified NTPC patterns: a marks-scheme word problem turned into a linear equation (June 2026 UG CBT-1) and a system of three linear equations in three variables (March 2026 Graduate CBT-1). Two-variable consistency and inequality questions had no verified NTPC shift and are tagged syllabus-based.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Word statement to one linear equation (marking scheme type)
[PYQ: NTPC UG CBT-1 13-Jun-2026 Shift-3]
EXAM LEVEL
Q. A test has 60 questions. Each correct answer earns 3 marks and each wrong answer loses 1 mark. A candidate attempts all and scores 120. How many did he get right?
Let c be correct, so 60 − c are wrong. 3c − (60 − c) = 120 gives 4c = 180 and c = 45. Formula check: (120 + 1 × 60)/(3 + 1) = 45.
Answer: 45
EXAMATLAS LEVEL
Q. A test has 80 questions, with +4 for a correct answer and −1 for a wrong one. A candidate leaves 10 questions and scores 180. Find the number of correct and wrong answers.
He attempts 70, so c + w = 70 and 4c − w = 180. Adding, 5c = 250 and c = 50, so w = 20. Check: 200 − 20 = 180.
Answer: 50 correct, 20 wrong
Pattern 2: Two equations in two variables and their consistency
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. Solve 3x + 2y = 16 and 2x − y = 6.
From the second, y = 2x − 6. Substituting, 3x + 4x − 12 = 16, so 7x = 28, x = 4 and y = 2.
Answer: x = 4, y = 2
EXAMATLAS LEVEL
Q. For what value of k do kx + 3y = 6 and 4x + 6y = 9 have no solution? For what values of k and m do 2x + 3y = 7 and kx + 6y = m have infinitely many solutions?
No solution needs k/4 = 3/6 with 3/6 ≠ 6/9; k = 2 works since 1/2 ≠ 2/3. Infinitely many needs 2/k = 3/6 = 7/m, so k = 4 and m = 14.
Answer: k = 2; k = 4 and m = 14
Pattern 3: Three equations in three variables
[PYQ: NTPC Graduate CBT-1 25-Mar-2026 Shift-1]
EXAM LEVEL
Q. Solve x + y + z = 6, x − y + z = 2 and x + y − z = 0.
Subtracting the second from the first, 2y = 4, so y = 2. Subtracting the third from the first, 2z = 6, so z = 3. Then x = 6 − 2 − 3 = 1.
Answer: x = 1, y = 2, z = 3
EXAMATLAS LEVEL
Q. Solve 2x + 3y − z = 5, x − y + 2z = 5 and 3x + y + z = 9, and find xyz.
Adding the first and third removes z: 5x + 4y = 14. Doubling the first and adding the second removes z: 5x + 5y = 15, so x + y = 3. Subtracting 5x + 4y = 14 from 5x + 5y = 15 gives y = 1, so x = 2. From the third, z = 9 − 6 − 1 = 2. Check in the second: 2 − 1 + 4 = 5. xyz = 4.
Answer: x = 2, y = 1, z = 2; xyz = 4
Pattern 4: Linear inequalities and counting integer solutions
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. How many integers satisfy both 3x − 5 < 2x + 4 and 2x + 1 ≥ 7?
The first gives x < 9 and the second gives x ≥ 3. Integers 3, 4, 5, 6, 7, 8: six of them.
Answer: 6
EXAMATLAS LEVEL
Q. How many integers x satisfy −3 ≤ (2x − 1)/3 < 5? Also solve 5 − 2x > 11.
Multiplying by 3, −9 ≤ 2x − 1 < 15, so −8 ≤ 2x < 16 and −4 ≤ x < 8: integers −4 to 7, which is 12. For the second, −2x > 6, and dividing by −2 reverses the sign: x < −3.
Answer: 12 integers; x < −3
CBT-2 LEVEL
Pattern 5: Equations in 1/x and 1/y (reciprocal substitution)
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. Solve 2/x + 3/y = 13 and 5/x − 4/y = −2.
Put u = 1/x and v = 1/y: 2u + 3v = 13 and 5u − 4v = −2. Multiplying the first by 4 and the second by 3 and adding, 23u = 46, so u = 2 and v = 3. Hence x = 1/2 and y = 1/3.
Answer: x = 1/2, y = 1/3
EXAMATLAS LEVEL
Q. Solve 6/(x + y) + 4/(x − y) = 4 and 9/(x + y) − 2/(x − y) = 2.
Put p = 1/(x + y) and q = 1/(x − y): 6p + 4q = 4 and 9p − 2q = 2. Doubling the second and adding, 24p = 8, so p = 1/3 and q = (4 − 2)/4 = 1/2. Then x + y = 3 and x − y = 2, giving x = 5/2 and y = 1/2.
Answer: x = 5/2, y = 1/2
60-Second Revision
- Turn the statement into one equation; check the answer back in the statement.
- Unique, none or infinite solutions: compare a₁/a₂, b₁/b₂ and c₁/c₂.
- Three variables: add or subtract pairs to kill one variable at a time.
- Dividing an inequality by a negative number reverses it.
- Reciprocal equations become linear with u = 1/x, v = 1/y.