Linear Equations in Two Variables Revision Notes - Class 10 Mathematics

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📌 Key Points

Linear equation in two variables: ax + by + c = 0 where a, b ≠ 0
A linear equation in two variables has infinitely many solutions
When plotted, all solutions of a linear equation form a straight line
x-intercept: Put y = 0 and solve for x
y-intercept: Put x = 0 and solve for y
For unique solution: a₁/a₂ ≠ b₁/b₂ (lines intersect at one point)
For infinitely many solutions: a₁/a₂ = b₁/b₂ = c₁/c₂ (same line)
For no solution (inconsistent): a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (parallel lines)
Substitution method: Express one variable in terms of other, then substitute
Elimination method: Make coefficients equal and add/subtract to eliminate variable
Cross-multiplication: Direct formula for solution without intermediate steps
Graphical method: Plot both lines and find their point of intersection
Parallel lines: Same slope (a₁/a₂ = b₁/b₂) but different y-intercepts
Coincident lines: One equation is a multiple of the other (all ratios equal)
Word problems: Identify two unknowns, form equations, solve algebraically
Always verify solution by substituting in both original equations
For consistent system: Has at least one solution (unique or infinite)
For inconsistent system: No solution (parallel lines never meet)
Dependent system: Infinitely many solutions (equations represent same line)
Independent system: Equations represent different lines with unique solution

📘 Important Definitions

Linear Equation in Two Variables

An equation of the form ax + by + c = 0 where a, b, c are real numbers and a, b ≠ 0.

Solution of Linear Equation

Any pair (x, y) of real numbers that satisfies the equation.

Standard Form

The form ax + by + c = 0 is called standard form. Other forms: y = mx + c (slope-intercept), x/a + y/b = 1 (intercept form).

Consistent System

A system of equations that has at least one solution (unique or infinite).

Inconsistent System

A system of equations that has no solution. Lines are parallel.

Dependent System

A system with infinitely many solutions. Equations represent the same line.

Independent System

A system where equations are different. Unique solution exists.

x-intercept

The point where the line crosses the x-axis. Found by setting y = 0.

y-intercept

The point where the line crosses the y-axis. Found by setting x = 0.

Substitution Method

Expressing one variable in terms of the other, then substituting in the second equation.

Elimination Method

Making coefficients of one variable equal and adding/subtracting equations to eliminate that variable.

Cross-Multiplication Method

Using the formula x/(b₁c₂ - b₂c₁) = y/(c₁a₂ - c₂a₁) = 1/(a₁b₂ - a₂b₁) for direct solution.

🔢 Formulas & Laws

Standard Form

ax + by + c = 0

Slope-Intercept Form

y = mx + c, where m is slope and c is y-intercept

Intercept Form

x/a + y/b = 1, where a is x-intercept and b is y-intercept

Consistency Conditions

Unique solution: a₁/a₂ ≠ b₁/b₂ | Infinite solutions: a₁/a₂ = b₁/b₂ = c₁/c₂ | No solution: a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Cross-Multiplication Formula

x/(b₁c₂ - b₂c₁) = y/(c₁a₂ - c₂a₁) = 1/(a₁b₂ - a₂b₁)

⚠️ Common Mistakes

✗ Wrong: Confusing linear with quadratic equations

✓ Correct: Linear equations have variables with power 1 only. Quadratic has x² or y².

✗ Wrong: Wrong intercepts

✓ Correct: For x-intercept, set y = 0 (not x = 0). For y-intercept, set x = 0 (not y = 0).

✗ Wrong: Incorrect consistency check

✓ Correct: All three ratios must be compared. If first two equal but third different → no solution (inconsistent).

✗ Wrong: Sign errors in elimination method

✓ Correct: When multiplying equations, multiply ALL terms including the constant. Check signs carefully.

✗ Wrong: Not verifying solutions

✓ Correct: Always substitute the solution back in BOTH original equations to verify.

✗ Wrong: Forgetting to simplify fractions

✓ Correct: Always reduce solutions to simplest form. If solution is (9/8, 13/8), leave in fraction form unless asked for decimal.

✗ Wrong: Incorrect word problem setup

✓ Correct: Read problem carefully. 'Sum' means addition, 'difference' means subtraction, 'times' means multiplication.

✗ Wrong: Choosing wrong variable for substitution

✓ Correct: Choose the variable with coefficient 1 or -1 for easier substitution.

📝 Exam Focus

These questions are frequently asked in CBSE exams:

Consistency of systems - most frequently asked concept

All three solving methods - substitution, elimination, cross-multiplication

Word problems involving two unknowns

Graphical representation and finding intersection points

Verification of solutions

Identifying parallel, intersecting, and coincident lines

Finding x and y intercepts

Determining whether a point lies on a line

🎯 Last-Minute Recall