Pair of Linear Equations in Two Variables

1. Chapter at a glance

• A pair of linear equations in two variables can be solved graphically or algebraically.
• Graphically, the lines may intersect at one point (unique solution), be parallel (no solution) or coincide (infinitely many solutions).
• A pair with a unique solution is consistent; a pair with no solution is inconsistent; a pair with infinitely many solutions is dependent (and consistent).
• The nature of solutions is decided by comparing the ratios \( \frac{a_1}{a_2} \), \( \frac{b_1}{b_2} \) and \( \frac{c_1}{c_2} \).
• Substitution method: express one variable from one equation and substitute in the other.
• Elimination method: make coefficients of one variable equal and add or subtract the equations.
• Word problems are first translated into a pair of linear equations and then solved by any of the above methods.

2. Definitions, theorems and results

• A pair of linear equations which has no solution is called an inconsistent pair of linear equations.
• A pair of linear equations in two variables which has a solution is called a consistent pair of linear equations.
• A pair of linear equations which are equivalent (have infinitely many distinct common solutions) is called a dependent pair of linear equations; a dependent pair is always consistent.
• For the equations \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \):
• If \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \), the lines intersect (unique solution, consistent).
• If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \), the lines coincide (infinitely many solutions, dependent and consistent).
• If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \), the lines are parallel (no solution, inconsistent).
• The converse of the above statements is also true. (No proof required in the exam as per chapter content.)

3. Formula sheet

4. Solved-example patterns

• Graphical consistency check: Plot both lines by finding at least two points each; observe whether they intersect (unique solution), coincide or are parallel. Read the intersection point if it exists.
• Substitution method: (i) Express one variable in terms of the other from any convenient equation. (ii) Substitute in the second equation and solve for the remaining variable. (iii) Back-substitute to find the other variable. Check whether the resulting statement is true (infinitely many solutions) or false (no solution).
• Elimination method: (i) Multiply equations by suitable constants to make one pair of coefficients equal. (ii) Add or subtract to eliminate one variable. (iii) Solve the resulting linear equation. (iv) Substitute back to obtain the second variable. A true statement with no variable means infinitely many solutions; a false statement means no solution.
• Word problems: Translate the given conditions into two linear equations, solve by substitution or elimination, and verify that the values satisfy the original problem statements.
• Special cases (e.g., two-digit numbers): Form equations using place-value expressions; consider both possibilities when a difference condition is given (two sub-cases).

5. Common mistakes and exam pitfalls

• Forgetting to consider both cases when digits differ by 2 (x – y = 2 and y – x = 2) leads to missing one valid number.
• Sign errors while substituting or while adding/subtracting after making coefficients equal.
• Writing the ratios \( \frac{a_1}{a_2} \), \( \frac{b_1}{b_2} \), \( \frac{c_1}{c_2} \) incorrectly or comparing only two ratios instead of all three.
• Not verifying the obtained solution in the original word-problem statements (age, cost, fraction conditions).
• Assuming a unique solution exists without first checking the ratios; missing the “infinitely many / no solution” conclusions when the statement after elimination is an identity or contradiction.