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Linear Equations

BSF Constable · Quantitative Aptitude

Also prepare for:CRPF Const CISF Const ITBP Const UP Const

Quadratic Equations →

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Core Concept

Read this first — the foundation of the topic

→Core Concept

Linear equations represent straight lines when plotted on a graph. In SSC CGL, you'll mostly deal with one variable (like x) or two variables (like x and y). The key is finding the value of unknown variables

💡Key Rules

For one variable linear equations like ax + b = 0, the solution is x = -b/a. For two variable systems, you need two equations to find unique solutions. Always maintain balance - whatever you do to one side, do to the other side.

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Formula Block

Memorise — at least one formula appears in every paper

• One variable: ax + b = 0, solution x = -b/a

• Two variables: a1x + b1y = c1 and a2x + b2y = c2

• Elimination method: Multiply equations to make coefficients equal

• Substitution method: Express one variable in terms of another

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Exam Patterns

What examiners ask — read before attempting PYQs

SSC CGL loves testing linear equations through word problems. Age problems, mixture problems, and number problems frequently appear. Questions often involve finding two numbers given their sum and difference, or determining speeds and distances. Expect 2-3 questions per paper.

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Shortcuts

Use these to save 30–60 seconds per question

For sum-difference problems, use this lightning method. If sum = S and difference = D, then larger number = (S+D)/2 and smaller number = (S-D)/2. This skips the entire equation-solving process.

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Worked Example

Solve this step-by-step before moving on

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Step 1

Let the numbers be x and y where x > y

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Step 2

Given equations are x + y = 50 and x - y = 12

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Step 3

Using shortcut - Larger number = (50+12)/2 = 31

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Step 4

Smaller number = (50-12)/2 = 19

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Step 5

Verification: 31 + 19 = 50 ✓ and 31 - 19 = 12 ✓ Answer: The numbers are 31 and 19. Another Trick: For age problems, always define variables for present ages. If the problem mentions 'after n years' or 'before n years', add or subtract n from present ages respectively. Common Mistake: Students often confuse the setup in word problems. Read carefully whether the problem asks for present age or future age. Also, many forget to verify their answers by substituting back into original equations. Always cross-check your solutions to avoid silly errors that cost marks in competitive exams.

🔑 Key Points

Linear equation has highest power of variable as 1
For ax + b = 0, solution is x = -b/a
Two equations needed to solve two unknown variables
Sum-difference shortcut: larger = (S+D)/2, smaller = (S-D)/2
Age problems use present age as base, then add/subtract years
Elimination method makes coefficients equal before adding/subtracting
Substitution method expresses one variable through another
Always verify final answer by substituting back

📌 Exam Facts

Linear equation in one variable has exactly one solution
System of two linear equations can have unique solution, no solution, or infinite solutions
Slope-intercept form is y = mx + c where m is slope and c is y-intercept
Two parallel lines have same slope but different y-intercepts
Cramer's rule uses determinants to solve linear equation systems
Inconsistent system occurs when equations are contradictory
Dependent system has infinite solutions when equations are equivalent

🚀 60-Second Revision

Formula: x = -b/a for ax + b = 0
Shortcut: Sum-difference problems use (S±D)/2
Remember: Two equations solve two unknowns
Trap: Check if answer satisfies original equations
Method: Elimination for equal coefficients, substitution otherwise
Age rule: Present age ± years = required age
Verify: Always substitute final values back into equations

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Test yourself under real exam conditions

A timed BSF Constable mock shows exactly how Linear Equations questions appear in the actual paper — and where you lose marks.

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