Introduction to Linear Equations (2 Variables)
Introduction to linear equations in two variables.
What Are Linear Equations in Two Variables?
In earlier classes, you solved equations that involved only one unknown, such as 5x = 40. This chapter introduces a new and very useful idea: equations that involve two unknowns at the same time, such as 5x + 4y = 32. These are called linear equations in two variables, and they form the foundation for graphing straight lines, solving pairs of equations (simultaneous equations), and understanding real-world situations where two quantities are related.
The word "linear" simply means that every variable in the equation has power (exponent) 1 — there are no terms like x², y², or xy. As long as both x and y appear with power 1, the equation is linear, no matter how many numbers or fractions are attached to them.
x and y) and the highest power of each variable is 1, it is a linear equation in two variables. It can always be rearranged into the standard form ax + by + c = 0.
Quick Recap — Linear Equations in One Variable
Before jumping into two variables, let's recall what a linear equation in one variable looks like, using a simple everyday situation.
Let the cost of each balloon be x rupees. Since Ajay bought 5 balloons, the total cost is 5x. We are told this total equals ₹40, so we can write the equation:
So the cost of each balloon is ₹8. Notice that this equation, 5x = 40, contains only one variable — x — and its power is 1. This is exactly what we call a linear equation in one variable.
Other examples of linear equations in one variable include equations like 3p − 5 = 0 (variable p), t/3 + 7 = 2 (variable t), 3 = 5 − 3y (variable y), and √3 l − √7 = 4 (variable l). In every case, there is only one letter standing for an unknown number, and that letter's power is 1.
From One Variable to Two — Hari's Pens & Pencils
Now consider a slightly different situation. Hari bought 5 pens and 4 pencils for ₹32. Can you find the cost of each item? Unlike the balloon problem, here you have two unknowns — the price of a pen and the price of a pencil — so a single equation cannot be solved for a unique pair of values. What it can do is express the relationship between the two unknown prices.
The resulting equation 5x + 4y = 32 has two variables (x and y), each with power 1. That is exactly the definition of a linear equation in two variables.
5x + 4y = 32, there are many possible pairs of values for (x, y) that satisfy it — for example, if x = 4 then y = 3, or if x = 0 then y = 8. A single linear equation in two variables does not have one unique solution — it has infinitely many. That is why, to find the exact price, we would need a second equation (a system / pair of equations).
The General Form — ax + by + c = 0
Every linear equation in two variables can be rewritten in the standard (general) form. This is a single template that covers all possible linear equations in two variables, no matter how they look at first glance.
ax + by + c = 0
| Symbol | What It Represents | Restriction |
|---|---|---|
| a | Coefficient of x (the number multiplied with x) | Any real number; a and b cannot both be zero at the same time |
| b | Coefficient of y (the number multiplied with y) | Any real number; same restriction as above |
| c | Constant term (a plain number with no variable) | Any real number, including 0 |
| x, y | The two variables (unknowns) | Each appears with power exactly 1 |
0·x + 0·y + c = 0, which simplifies to just c = 0 — a statement about a constant, not an equation with variables at all. So we always require that at least one of a, b is non-zero.
Let's look at several examples of equations in two variables and confirm they fit the general form:
a, b, and c can be whole numbers, fractions, decimals, or even surds like √5 — any real number is allowed. What matters is that x and y appear with power 1.
Worked Examples — Expressing Equations in General Form
A common board exam task is: "Write the following equation in the form ax + by + c = 0 and state the values of a, b, c." The method is simple — move all terms to the left-hand side so the right-hand side becomes zero, then read off the coefficients.
LHS − RHS = 0. Whatever is left on the left side, arrange as ax + by + c and read off a, b, c directly.
Here, the coefficient of x is negative (−2). This is perfectly valid — coefficients can be any real number, positive or negative.
This equation is already equal to zero on the right side — so it is already in general form! Just identify the coefficients:
Here, the right-hand side is just y, not a number. Move y to the left side:
Notice that c = 0 here, because there is no constant term. The equation still qualifies as a linear equation in two variables because a and b are not both zero.
Quick Reference — All "Try This" Answers at a Glance
| Part | Original Equation | General Form | a | b | c |
|---|---|---|---|---|---|
| (i) | 3x + 2y = 9 | 3x + 2y − 9 = 0 | 3 | 2 | −9 |
| (ii) | −2x + 3y = 6 | −2x + 3y − 6 = 0 | −2 | 3 | −6 |
| (iii) | 9x − 5y = 10 | 9x − 5y − 10 = 0 | 9 | −5 | −10 |
| (iv) | x/2 − y/3 − 5 = 0 | Already in general form | 1/2 | −1/3 | −5 |
| (v) | 2x = y | 2x − y + 0 = 0 | 2 | −1 | 0 |
Common Mistakes to Avoid
- Forgetting to change the sign when moving a term: When you move a term from the right side to the left side, it changes sign. So if the equation is
3x + 2y = 9, moving 9 gives3x + 2y − 9 = 0(not+9). - Calling an equation linear when a variable has power ≠ 1: Equations like
x² + y = 5orxy = 6are not linear, because x² has power 2 and xy is a product of two variables. Always check the powers. - Thinking there is only one solution: A single linear equation in two variables has infinitely many solutions (one for every value you choose for x or y). Don't try to solve it for a unique pair without a second equation.
- Mixing up coefficients: When writing a = __, b = __, c = __, always pair a with x and b with y. Students sometimes swap them in exams.
- Dropping the negative sign on b: If the equation is
9x − 5y − 10 = 0, then b = −5 (negative!), not +5. Write the sign as part of the coefficient.
9x − 5y − 10 = 0. The minus sign belongs to the coefficient — always include it.
One Variable vs Two Variables — Side by Side
| Feature | Linear Equation — One Variable | Linear Equation — Two Variables |
|---|---|---|
| Form | ax + b = 0 | ax + by + c = 0 |
| Unknowns | 1 (e.g. only x) | 2 (e.g. x and y) |
| Power of variables | 1 | 1 (for both x and y) |
| Number of solutions | Exactly 1 unique solution | Infinitely many solutions |
| Example | 5x = 40 → x = 8 | 5x + 4y = 32 (many pairs work) |
| Graph | A single point on a number line | A straight line on the xy-plane |
What This Introduction Prepares You For
This introduction to linear equations in two variables is the first step in Chapter 6. Once you are comfortable with recognising and writing equations in the form ax + by + c = 0, the chapter moves on to finding solutions (pairs of values that satisfy the equation) and then to graphing those solutions as a straight line on the coordinate plane.
In Class 10, you will study pairs of linear equations in two variables — where two such equations are given simultaneously and you must find the single (x, y) pair that satisfies both. The methods used there — substitution, elimination, and graphical — all rest on the ideas introduced here. Mastering this chapter is also essential for Class 10's Coordinate Geometry and for Polynomials, where you interpret graphs of algebraic expressions.
For students appearing in the Telangana and Andhra Pradesh board exams, questions on Chapter 6 regularly appear as 2-mark or 4-mark problems asking you to: (a) express a word problem as a linear equation in two variables, (b) identify a, b, c from the general form, or (c) find solutions and plot the corresponding line. This introduction lays the conceptual groundwork for all three question types.
ax + by + c = 0 first, then read off the values. Never try to guess from the original equation directly — moving all terms to the left side first prevents sign mistakes.