Pair of Linear Equations in Two Variables — Study Notes

1. Introduction

A linear equation in two variables is an equation of the form ax + by + c = 0, where a, b, and c are real numbers and a and b are not both zero. A single such equation has infinitely many solutions, as each solution (x, y) corresponds to a point lying on a straight line. This chapter studies what happens when we consider two such equations together, that is, a pair of linear equations, and how to find their common solutions.

The pair of equations represents two straight lines in the plane. Their common solution is the point where the two lines intersect, if they do. Depending on the relative position of the lines, a system may have a unique solution, infinitely many solutions, or no solution at all. These three cases correspond to intersecting lines, coincident lines, and parallel lines respectively.

This chapter equips students with three algebraic methods to solve such systems: the substitution method, the elimination method, and the cross-multiplication method. These methods are used not only to solve pure mathematical problems but also to model and solve word problems involving ages, numbers, speeds, and mixtures, which form a significant part of the board examination.

2. Graphical Representation and Types of Solutions

Consider the pair a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0.

Case 1: Intersecting Lines (Unique Solution)

The lines intersect at exactly one point if: a1/a2 is not equal to b1/b2

Case 2: Coincident Lines (Infinitely Many Solutions)

The lines coincide if: a1/a2 = b1/b2 = c1/c2

Case 3: Parallel Lines (No Solution)

The lines are parallel and distinct if: a1/a2 = b1/b2 is not equal to c1/c2

A pair of linear equations is said to be consistent if it has at least one solution (Cases 1 and 2) and inconsistent if it has no solution (Case 3).

3. Substitution Method

Steps

1. From one of the equations, express one variable in terms of the other, e.g., express y in terms of x.
2. Substitute this expression into the other equation, converting it into a linear equation in one variable.
3. Solve for that variable.
4. Substitute the value back to find the other variable.

Example: Solve x + y = 5 and x - y = 1. From the first, y = 5 - x. Substituting into the second: x - (5 - x) = 1, so 2x - 5 = 1, x = 3. Then y = 2. Hence the solution is x = 3, y = 2.

The substitution method is especially convenient when one of the coefficients of a variable is 1, making the substitution straightforward.

4. Elimination Method

Steps

1. Multiply one or both equations by suitable constants so that the coefficients of one variable become numerically equal (with opposite signs if adding).
2. Add or subtract the equations to eliminate that variable.
3. Solve the resulting equation in one variable.
4. Substitute to find the other variable.

Example: Solve 3x + 2y = 11 and 2x + 3y = 4. Multiply the first by 2 and the second by 3: 6x + 4y = 22 and 6x + 9y = 12. Subtracting: -5y = 10, so y = -2. Then 3x - 4 = 11, so x = 5. Solution: x = 5, y = -2.

The elimination method is systematic and works well even when coefficients are not simple.

5. Cross-Multiplication Method

For the system a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, the solution is given by: x/(b1c2 - b2c1) = y/(c1a2 - c2a1) = 1/(a1b2 - a2b1)

Provided a1b2 - a2b1 is not zero. Equivalently: x = (b1c2 - b2c1)/(a1b2 - a2b1) y = (c1a2 - c2a1)/(a1b2 - a2b1)

The denominator a1b2 - a2b1 vanishes precisely when the lines are parallel or coincident, which is the same condition as a1/a2 = b1/b2.

Memory Aid

Write the coefficients in the array a1 b1 c1 / a2 b2 c2. Then x numerator = b1c2 - b2c1, y numerator = c1a2 - c2a1, and common denominator = a1b2 - a2b1, taking care with signs.

6. Equations Reducible to Linear Form

Some equations that do not look linear can be reduced to linear form by substitution. For example, the pair: 1/x + 1/y = 7 and 2/x + 3/y = 17 can be solved by letting u = 1/x and v = 1/y, giving the linear system u + v = 7 and 2u + 3v = 17.

Solving: multiply the first by 2 to get 2u + 2v = 14; subtract from the second: v = 3, then u = 4. Hence 1/x = 4 and 1/y = 3, so x = 1/4 and y = 1/3.

This technique is frequently applied to word problems involving reciprocals of quantities such as speeds of boats and streams or the time taken to do a piece of work.

7. Word Problems

Word problems translate real situations into a pair of linear equations. The steps are:

1. Read the problem carefully and identify the unknowns; assign variables x and y.
2. Translate each condition into an equation.
3. Solve the resulting system by the most convenient method.
4. Check the solution against the wording of the problem and write the final answer in a sentence.

Common contexts include ages, sums of digits of numbers, speed and distance, money and tickets, and mixing of solutions. Correct framing of equations is the most important skill tested here, and full marks usually depend on both the equations and the final verified answer.

Quick Revision Tables

Table 1: Conditions for the Three Cases

Table 2: Comparison of Solution Methods

Mind Map

Important Diagrams (SVG)

Diagram 1: The Three Possible Line Arrangements

Diagram 2: Elimination Method Flow

Common Mistakes

1. Getting the condition for parallel lines wrong: writing a1/a2 = b1/b2 = c1/c2 for parallel lines, which actually gives coincident lines. Parallel lines require c1/c2 to differ.
2. Forgetting that a and b in ax + by + c = 0 cannot both be zero; a linear equation must contain at least one variable.
3. In the substitution method, substituting the expression into the same equation it came from, leading to an identity instead of a solution.
4. Sign errors in cross-multiplication, especially mixing up b1c2 - b2c1 with c1b2 - c2b1.
5. In elimination, subtracting when the signs of coefficients are the same and adding when opposite, but doing the reverse.
6. Declaring a system inconsistent just because the lines are different, without actually checking the ratio condition.
7. In word problems, assigning x and y but writing equations that do not match the wording, e.g., reversing the sum of digits of a two-digit number.

Exam Tips

1. For ratio-based questions, always write a1/a2, b1/b2, and c1/c2 in the given order and compare carefully.
2. In elimination, if coefficients of x are positive in both equations, subtract; if one is negative, add. Check signs before operating.
3. Practise cross-multiplication by writing the coefficient array neatly; it saves time in 3-mark questions.
4. In word problems, define x and y clearly at the start with proper units, and verify your answer fits the story.
5. For equations like 1/x + 1/y = 7, remember to substitute back to recover x and y, and state the final solution.
6. Learn the shortcut: if a1b2 - a2b1 is not zero, there is a unique solution; if the determinant is zero, check further.
7. Always verify the solution in both equations before writing the final answer; a single verification mistake can cost full marks.

Conclusion

A pair of linear equations in two variables is a central topic that connects geometry and algebra. Understanding the conditions for intersecting, coincident, and parallel lines gives students an intuitive geometric feel for systems of equations, while the three algebraic methods provide reliable tools for computation. This chapter teaches a general problem-solving strategy of framing equations from real-world situations and solving them systematically, a skill that extends far beyond mathematics. Together with its frequent appearance in board exams, it lays the groundwork for linear programming and systems of equations studied in higher classes.