📈 Graphical Representation
Standard Form
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Each equation represents a straight line on the coordinate plane. The solution is the point(s) of intersection.
UNIQUE SOLUTION
Intersecting Lines
Lines cross at exactly one point. That point (x, y) is the unique solution. System is consistent & independent.
- a₁/a₂ ≠ b₁/b₂
- Example: x + y = 5, 2x − y = 1 → (2, 3)
NO SOLUTION
Parallel Lines
Lines never meet. No solution exists. System is inconsistent.
- a₁/a₂ = b₁/b₂ ≠ c₁/c₂
- Example: x + y = 5, x + y = 8
INFINITE SOLUTIONS
Coincident Lines
Same line — infinitely many solutions. System is consistent & dependent.
- a₁/a₂ = b₁/b₂ = c₁/c₂
- Example: x + y = 5, 2x + 2y = 10
🔢 Consistency Conditions
For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Compare ratios: a₁/a₂ vs b₁/b₂ vs c₁/c₂
Interactive Ratio Check
Equation 1: a₁x + b₁y + c₁ = 0
Equation 2: a₂x + b₂y + c₂ = 0
MEMORY TIP
The Ratio Rule
Think of it as a 3-way comparison: if the first ratio doesn't match the second, lines intersect (unique). If all three match, coincident. If first two match but third doesn't — parallel.
EXAM TRICK
Quick Test
Write ratios as fractions in lowest terms. Compare a₁/a₂ and b₁/b₂ first. If equal, check c₁/c₂. Three equal → coincident; two equal → parallel; first unequal → intersecting.
🔄 Substitution Method
Worked Example
2x + 3y = 11 and x − 2y = −1
Express one variable in terms of the other, then substitute into the second equation.
Step 1 / 8
WHEN TO USE
Best Situations
- One equation has coefficient of 1 or −1
- Easy to isolate a variable without fractions
- Small coefficients
STEPS
Algorithm
- Express x (or y) from one equation
- Substitute into second equation
- Solve for the remaining variable
- Back-substitute to find the other
COMMON ERRORS
Watch Out For
- Sign errors when substituting negative terms
- Forgetting to distribute multiplied terms
- Not verifying the answer in both equations
➕ Elimination Method
Worked Example
3x + 2y = 16 and 5x − 3y = 7
Make coefficients of one variable equal (with opposite signs), then add equations to eliminate it.
Step 1 / 8
WHEN TO USE
Best Situations
- Coefficients are larger numbers
- Both equations have the same variable form
- Multiplying is neater than isolating
STEPS
Algorithm
- Multiply each equation by suitable constants
- Make LCM of one variable's coefficients equal
- Add or subtract equations to eliminate
- Solve for remaining variable; back-substitute
KEY INSIGHT
Add vs Subtract
If equal coefficients have opposite signs → add equations.
If same signs → subtract. Both eliminate the chosen variable.
✖ Cross-Multiplication Method
Formula — for a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
x/(b₁c₂ − b₂c₁) = y/(c₁a₂ − c₂a₁) = 1/(a₁b₂ − a₂b₁)
Condition: a₁b₂ − a₂b₁ ≠ 0 (otherwise lines are parallel or coincident)
Butterfly / Arrow Diagram
a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0
a₂x + b₂y + c₂ = 0
Write coefficients in order: b, c, a, b (repeat first column at end)
b₁c₁a₁b₁
b₂c₂a₂b₂
↕ cross ↕ ↕ cross ↕
b₁c₂ − b₂c₁
= x's numerator
value of x = (b₁c₂ − b₂c₁) ÷ (a₁b₂ − a₂b₁)
c₁a₂ − c₂a₁
= y's numerator
value of y = (c₁a₂ − c₂a₁) ÷ (a₁b₂ − a₂b₁)
a₁b₂ − a₂b₁
= common denominator
must be ≠ 0 for unique solution
Worked Example — 2x + 3y + 5 = 0 and 3x − 2y + 1 = 0
x / (3×1 − (−2)×5) = y / (5×3 − 1×2) = 1 / (2×(−2) − 3×3)
x / (3 + 10) = y / (15 − 2) = 1 / (−4 − 9)
x / 13 = y / 13 = 1 / (−13)
x = −1, y = −1
x / (3 + 10) = y / (15 − 2) = 1 / (−4 − 9)
x / 13 = y / 13 = 1 / (−13)
x = −1, y = −1
WHEN TO USE
Best For
- Direct formula application in exams
- Competitive exam problems needing speed
- When coefficients are messy fractions
COMMON ERROR
Watch Sign
The formula requires equations in the form ax + by + c = 0 (not ax + by = c). Move all terms to one side before applying the cross-product rule.
🎯 Problem Types & Applications
Click any category to see a solved example setup.
Age Problems
Set up equations using present and future/past ages. Always define clearly: let father's age = x, son's age = y.
Example: Father is 3× older than son. In 12 years, he'll be 2× older.
Let father = x, son = y
Eq 1: x = 3y
Eq 2: x + 12 = 2(y + 12)
Solving: y = 12, x = 36
✓ Father: 36, Son: 12
Let father = x, son = y
Eq 1: x = 3y
Eq 2: x + 12 = 2(y + 12)
Solving: y = 12, x = 36
✓ Father: 36, Son: 12
Money & Cost Problems
Price × quantity = total cost. Create two equations from two different purchase scenarios.
Example: 2 pencils + 3 pens = ₹18; 4 pencils + 5 pens = ₹32.
Let pencil = x, pen = y
Eq 1: 2x + 3y = 18
Eq 2: 4x + 5y = 32
Solving: x = 3, y = 4
✓ Pencil: ₹3, Pen: ₹4
Let pencil = x, pen = y
Eq 1: 2x + 3y = 18
Eq 2: 4x + 5y = 32
Solving: x = 3, y = 4
✓ Pencil: ₹3, Pen: ₹4
Speed, Distance & Time
Use D = S × T. With boats/streams: upstream speed = (u − v), downstream = (u + v).
Example: A boat takes 2h downstream (30 km) and 3h upstream (30 km).
Let boat speed = x, stream = y
Eq 1: x + y = 15 (downstream)
Eq 2: x − y = 10 (upstream)
Solving: x = 12.5, y = 2.5
✓ Boat: 12.5 km/h, Stream: 2.5 km/h
Let boat speed = x, stream = y
Eq 1: x + y = 15 (downstream)
Eq 2: x − y = 10 (upstream)
Solving: x = 12.5, y = 2.5
✓ Boat: 12.5 km/h, Stream: 2.5 km/h
Number & Digit Problems
Express two-digit numbers as 10a + b (tens digit a, units digit b). Use given clues to form equations.
Example: Sum of digits of a 2-digit number = 9. On reversing, number increases by 27.
Let tens = x, units = y
Eq 1: x + y = 9
Eq 2: (10y + x) − (10x + y) = 27 → y − x = 3
Solving: x = 3, y = 6
✓ Number = 36
Let tens = x, units = y
Eq 1: x + y = 9
Eq 2: (10y + x) − (10x + y) = 27 → y − x = 3
Solving: x = 3, y = 6
✓ Number = 36
Geometry Problems
Area, perimeter, angle sum properties of triangles/quadrilaterals often yield linear systems.
Example: In a triangle, one angle is 40° more than another. The third = 60°.
Let angles = x, y, z = 60°
Eq 1: x + y + 60 = 180 → x + y = 120
Eq 2: x − y = 40
Solving: x = 80°, y = 40°
✓ Angles: 80°, 40°, 60°
Let angles = x, y, z = 60°
Eq 1: x + y + 60 = 180 → x + y = 120
Eq 2: x − y = 40
Solving: x = 80°, y = 40°
✓ Angles: 80°, 40°, 60°
Fraction Problems
Add/subtract to numerator or denominator gives a new fraction. Let numerator = x, denominator = y.
Example: Adding 1 to numerator makes it 1/2; adding 1 to denominator makes it 1/3.
Let fraction = x/y
Eq 1: (x+1)/y = 1/2 → 2x − y = −2
Eq 2: x/(y+1) = 1/3 → 3x − y = 1
Solving: x = 3, y = 8
✓ Fraction = 3/8
Let fraction = x/y
Eq 1: (x+1)/y = 1/2 → 2x − y = −2
Eq 2: x/(y+1) = 1/3 → 3x − y = 1
Solving: x = 3, y = 8
✓ Fraction = 3/8
General Strategy for Word Problems
Step 1 →
Identify the two unknown quantities and define them clearly (let x = ..., y = ...)
Identify the two unknown quantities and define them clearly (let x = ..., y = ...)
Step 2 →
Translate each condition in the problem into a linear equation
Translate each condition in the problem into a linear equation
Step 3 →
Choose the most suitable method (substitution / elimination) and solve
Choose the most suitable method (substitution / elimination) and solve
Step 4 →
Verify: substitute back into BOTH original equations. Check if answer makes sense
Verify: substitute back into BOTH original equations. Check if answer makes sense