Doubtclip is an AI tutor that turns any homework or exam doubt into a step-by-step whiteboard video lesson with diagrams, formulas, and voice explanation.

Who is the founder of Doubtclip?

Vinay Malav is the founder of Doubtclip (Founder, Doubtclip, based in Kota, Rajasthan, India). More: https://www.doubtclip.com/about

Which subjects and exams does it support?

It supports Physics, Chemistry, Mathematics/Calculus and related STEM doubts for school boards, JEE, NEET, and global high-school / undergrad practice.

How do I get a video solution?

Upload a photo of the question or paste the text, sign in, and Doubtclip generates a whiteboard-style explanation video you can watch and share.

Is Doubtclip available worldwide?

Yes. Doubtclip is a web app for students globally. Lessons are available in English and Hinglish.

The lines representing the pair of equations 2x + 4y = 9 and 3x + 6y = 12 are:

Parallel. a₁/a₂ = 2/3; b₁/b₂ = 4/6 = 2/3; c₁/c₂ = 9/12 = 3/4. Since a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the lines are parallel and have no solution.

Mathematics · Class 9-10 · CBSE

Pair of Linear Equations: Graphical and Algebraic Methods

Master systems of two linear equations: graphical representation, consistency conditions (unique, infinite, no solution), substitution and elimination methods, and word problems on ages, digits, and speed.

1System of Linear Equations and Consistency Conditions

A pair of linear equations in two variables x and y has the standard form:
a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0
where a₁, b₁, c₁, a₂, b₂, c₂ are real numbers, and a₁² + b₁² ≠ 0, a₂² + b₂² ≠ 0.

Consistency Conditions and Geometric Interpretations

Ratio Comparison	Graphical Representation	Algebraic Solution	System Consistency
a₁/a₂ ≠ b₁/b₂	Intersecting lines (cross at 1 point)	Exactly ONE Unique Solution	Consistent System
a₁/a₂ = b₁/b₂ = c₁/c₂	Coincident lines (coincide everywhere)	Infinitely Many Solutions	Consistent (Dependent) System
a₁/a₂ = b₁/b₂ ≠ c₁/c₂	Parallel lines (never intersect)	NO Solution	Inconsistent System

a₁/a₂ ≠ b₁/b₂
Graphical Representation
Intersecting lines (cross at 1 point)
Algebraic Solution
Exactly ONE Unique Solution
System Consistency
Consistent System
a₁/a₂ = b₁/b₂ = c₁/c₂
Graphical Representation
Coincident lines (coincide everywhere)
Algebraic Solution
Infinitely Many Solutions
System Consistency
Consistent (Dependent) System
a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Graphical Representation
Parallel lines (never intersect)
Algebraic Solution
NO Solution
System Consistency
Inconsistent System

2Algebraic Solving Methods: Substitution and Elimination

Elimination Method Step-by-Step Procedure

1
1. Multiply one or both equations by suitable non-zero constants so that coefficients of one variable (x or y) become equal in magnitude.
2
2. Add or subtract the two equations to eliminate that variable, leaving an equation in a single variable.
3
3. Solve the resulting linear equation to obtain the value of the remaining variable.
4
4. Substitute this calculated value into any original equation to solve for the other variable.

Real-life example · Worked Algebraic Elimination Example

$\text{Solve}\ : 2x + 3y = 11 \text{and}\ 2x - 4y = -24.$
• Subtract equation 2 from equation 1: (2x - 2x) + (3y - (-4y)) = 11 - (-24)
• 7y = 35 ⇒ y = 5.
• Substitute y = 5 into first equation: 2x + 3(5) = 11 ⇒ 2x + 15 = 11 ⇒ 2x = -4 ⇒ x = -2.
• Solution: (x, y) = (-2, 5).

3Standard Word Problems on Upstream/Downstream and Ages

Standard Modeling Templates for Board Word Problems

Problem Type	Variables Assigned	Standard Governing Equation
Boat & River Stream	x = boat speed in still water, y = stream speed	Downstream speed = x + y; Upstream speed = x - y. Time = Distance / Speed
Two-digit Number	x = tens digit, y = units digit	Original Number = 10x + y; Reversed Number = 10y + x
Age Problems	x = parent age now, y = child age now	n years ago: (x - n) and (y - n); in n years: (x + n) and (y + n)

Boat & River Stream
Variables Assigned
x = boat speed in still water, y = stream speed
Standard Governing Equation
Downstream speed = x + y; Upstream speed = x - y. Time = Distance / Speed
Two-digit Number
Variables Assigned
x = tens digit, y = units digit
Standard Governing Equation
Original Number = 10x + y; Reversed Number = 10y + x
Age Problems
Variables Assigned
x = parent age now, y = child age now
Standard Governing Equation
n years ago: (x - n) and (y - n); in n years: (x + n) and (y + n)

4Exam Corner: Tips, Mistakes & Board Questions

High-Yield Board Revision Takeaways

Always ensure both equations are written in standard form (both constants on the right side, or both on the left side) before computing ratio c₁/c₂ to avoid sign errors.
In boat problems, the speed of the boat in still water (x) must always be strictly greater than the speed of the water stream (y), otherwise upstream speed (x - y) would be negative.
For unique solution, ratio c₁/c₂ is not required; only a₁/a₂ ≠ b₁/b₂ is tested.

Exam question · CBSE Class 10 Board

For what value of k will the system of equations kx + 3y = k - 3 and 12x + ky = k have infinitely many solutions?

A. $k = 6$
B. $k = -6$
C. $k = 0$
D. $k = 3$

Remember

Intersecting linesa₁/a₂ ≠ b₁/b₂have exactly one unique solution.
Parallel lines ==a₁/a₂ = b₁/b₂ ≠ c₁/c₂== have no solution (inconsistent).
Coincident lines ==a₁/a₂ = b₁/b₂ = c₁/c₂== have infinitely many solutions.
Boat problems: Downstream speed isx + y; Upstream speed isx - y.

Check yourself

Question 1
The lines representing the pair of equations 2x + 4y = 9 and 3x + 6y = 12 are: