This page covers Agniveer Army CEE Linear Equations with complete concept notes, 8 graded practice MCQs, key points and exam-specific tips. Free to study.
Core ConceptRead 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.
Formula BlockMemorise — 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
Exam PatternsWhat 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.
ShortcutsUse 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.
Worked ExampleSolve this step-by-step before moving on
1
Step 1
Let the numbers be x and y where x > y
2
Step 2
Given equations are x + y = 50 and x - y = 12
3
Step 3
Using shortcut - Larger number = (50+12)/2 = 31
4
Step 4
Smaller number = (50-12)/2 = 19
5
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.
Exam TrapsCommon mistakes students make — avoid these
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 to Remember
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
A number when multiplied by 4 and then increased by 7 equals 35. Find the number.
Practice 3easy
The sum of two consecutive numbers is 49. Find the smaller number.
Practice 4medium
A soldier's current age is 5 years less than twice his brother's age. If his brother is 12 years old, what is the soldier's current age?
Practice 5medium
The sum of three consecutive integers is 84. What is the smallest of the three integers?
Practice 6medium
A shopkeeper sells books at ₹40 each. After selling some books, he earns ₹320. How many books did he sell?
Practice 7medium
If 3x + 7 = 22, what is the value of x?
Practice 8hard
A shopkeeper buys pens at ₹5 each and sells them at ₹8 each. After selling some pens, he makes a profit of ₹60. If he also gives a discount of ₹2 per pen to bulk buyers, and after the discount his profit per pen becomes ₹1, how many pens did he sell at the discounted rate to make a total profit of ₹60?
60-Second Revision — Linear Equations
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