Recap — Linear Equations in One and Two Variables

A linear equation is an equation whose graph is a straight line. Before working with pairs, let us distinguish between equations in one variable and two variables.

Try This — Identifying Linear Equations (MCQ Practice)

The lesson includes five multiple-choice questions to check understanding of what makes an equation linear, how many solutions a two-variable linear equation has, and how to verify a solution.

Graphical Method of Finding Solutions

To solve a pair of linear equations graphically, we plot both equations as straight lines on the same coordinate plane and look at how the two lines relate to each other. The point of intersection (if any) gives the solution.

For each equation, we build a table of (x, y) values by choosing at least 3 values of x, computing the corresponding y, and plotting the points. Connecting them gives the line.

How to build a points table

Case 1 — Intersecting Lines (Unique Solution)

Equations: 3x + y = 7   and   2x − y = 3

When both lines are plotted, they cross at (2, 1). This is the unique solution.

Case 2 — Parallel Lines (No Solution)

Equations: 3x + y = 10   and   6x + 2y = 24

Notice that the second equation (6x + 2y = 24) simplifies to 3x + y = 12 — it has the same left-hand side coefficients as the first equation but a different constant. These lines have the same slope but different y-intercepts, so they never meet.

Case 3 — Coincident Lines (Infinitely Many Solutions)

Equations: 2x + y = 3   and   6x + 3y = 9

The second equation is simply 3 times the first: 3 × (2x + y) = 3 × 3 = 9. They represent the same line. Every point that lies on one line also lies on the other.

The Three Cases — Visual Comparison

The Master Ratio Conditions Table

For any pair of linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, the nature of the solution depends entirely on the ratios of corresponding coefficients. This table is the most important thing to memorise in Chapter 4.

Try This — Worked Problems on Ratio Conditions

Common Mistakes to Avoid

• Writing equations in ax + by = c form vs ax + by + c = 0: The general form has c on the left side with a positive sign. If your equation is 2x + y = 7, then in standard form it is 2x + y − 7 = 0, making c = −7, not +7. Incorrect sign of c will give wrong ratio comparisons.
• Confusing c₁/c₂ for the parallel condition: For parallel lines, we need a₁/a₂ = b₁/b₂ BUT c₁/c₂ must be DIFFERENT. Students sometimes check all three ratios and declare coincident when c₁/c₂ also happens to match — always verify carefully.
• Not verifying the ≠ part for parallel lines (Try This 2): After finding k = −3 in Q2, you must check that c₁/c₂ (= 3/−5 = −3/5) is indeed ≠ a₁/a₂ (= 1/2). If both equalities held, the lines would be coincident, not parallel.
• Losing the p = 0 solution in Try This 4: The quadratic p(p − 6) = 0 gives two roots. Always mention both, then justify why the unwanted one is rejected.
• Wrong table values for graphing: Choose x-values that give integer or easy y-values to keep the graph accurate. Messy decimals on a graph make intersection points hard to read precisely.

What This Lesson Prepares You For

The Introduction establishes the graphical foundation for the three algebraic methods of solving a pair of linear equations taught in the subsequent exercises: Exercise 4.1 (Substitution Method), Exercise 4.2 (Elimination Method), and Exercise 4.3 (Cross Multiplication).

The ratio conditions (a₁/a₂, b₁/b₂, c₁/c₂) appear again in Class 10 Chapter 6 (Similar Triangles) when comparing corresponding side ratios, and in Class 11 Mathematics when studying system of linear equations using matrices and determinants.

Students who find this topic challenging should first review Class 9 Linear Equations in Two Variables, which covers how a single equation produces a line, and basic coordinate plotting.