Learn Solution of linear equations

In CBSE Class 9 Mathematics, the chapter on Linear Equations in Two Variables introduces you to algebraic relationships that represent real-world situations with two unknown quantities. Imagine you have ₹100 to spend on notebooks and pens; the number of notebooks and pens you can buy will vary, but their total cost remains fixed. This scenario can be expressed as a mathematical equation. A linear equation in two variables is written in the standard form ax + by + c = 0, where a, b, and c are real numbers, and a and b are not both zero. Finding a "solution" to this equation means discovering the specific pairs of values for the two variables that perfectly satisfy the condition.

The core logic behind the solution of linear equations lies in finding an ordered pair (x, y) that makes the left-hand side (LHS) of the equation equal to the right-hand side (RHS). Unlike linear equations in one variable, which have a single unique solution, a linear equation in two variables has infinitely many solutions. To calculate these solutions mathematically, you simply choose any random value for one variable (say, x), substitute it into the equation, and solve for the corresponding value of the other variable (y). Because you can pick an infinite number of values for x, you will continuously generate an infinite number of corresponding y values, creating an endless list of valid (x, y) coordinate pairs.

The diagram above illustrates both the algebraic and graphical approaches to finding the solutions of the linear equation x + y = 5. On the left, we use the substitution method to create a table of values. By picking random numbers for x (like 0, 2, and 5), we calculate the corresponding y values (5, 3, and 0), giving us the ordered pairs (0, 5), (2, 3), and (5, 0). On the right, these exact pairs are plotted on a Cartesian plane. When connected, these coordinate points form a perfectly straight line. In your school exams, you will frequently be asked to plot these lines; the golden rule to remember here is that every single point that lies on this straight line is a valid solution to the equation, beautifully proving why linear equations in two variables have an infinite number of solutions.

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