One equation, two unknowns → not one answer but a whole line of answers! Every linear equation in two variables draws a perfectly straight line on a graph.
Standard form
ax + by + c = 0, where a and b are not both zero.
Solution
A pair (x, y) that makes both sides equal — infinitely many exist.
Graph
Plot the solutions and they line up to form a straight line.
Axes
y = 0 is the x-axis; x = 0 is the y-axis.
1. What is a linear equation in two variables?
An equation of the form ax + by + c = 0 is called a linear equation in two variables, where a, b and c are real numbers and a and b are not both zero. Here x and y are the two variables. The word linear means the highest power of each variable is 1 — there is no x², no y², no xy, and no variable under a root. For example, 2x + 3y = 7, x − y = 0, and √2 x + y − 5 = 0 are all linear equations in two variables. An equation like 2x + 3 = 0 looks like a one-variable equation, but in this chapter we also treat it as a two-variable equation by writing it as 2x + 0·y + 3 = 0, so that a = 2, b = 0, c = 3.
2. Identifying a, b and c
To read off the coefficients, first bring the equation to standard form ax + by + c = 0. Take 3x + 4y = 12. Move 12 to the left: 3x + 4y − 12 = 0, so a = 3, b = 4, c = −12. Always shift every term to one side so the other side is 0 before identifying a, b and c. Watch the signs carefully — the constant keeps the sign it has after moving across the equals sign.
3. Solution of a linear equation
A solution is a pair of values, one for x and one for y, that satisfies the equation (makes the left side equal to the right side). We write a solution as an ordered pair (x, y), x first, y second. To find a solution, simply choose any value for one variable and solve for the other. For x + 2y = 6, choose x = 0 → 2y = 6 → y = 3, giving (0, 3). Choose x = 2 → 2y = 4 → y = 2, giving (2, 2). Choose y = 0 → x = 6, giving (6, 0). Because we can keep choosing different values, a linear equation in two variables has infinitely many solutions. This is the key difference from a linear equation in one variable, which has exactly one solution.
4. Why infinitely many?
With two variables we have one equation but two unknowns. One equation is not enough to pin down both values, so there is freedom: for every value we pick for x, the equation gives a matching y. Each such pair is a separate solution, and since there are infinitely many numbers to choose from, there are infinitely many solutions. To get a single unique solution we would need a second equation (studied later as a pair of linear equations).
5. Graph of a linear equation
If we plot all the solution pairs (x, y) of a linear equation on the Cartesian plane, every point lands exactly on one straight line. That is why these are called linear equations — their graph is a line. Conversely, every point on the line is a solution of the equation, and every point not on the line is not a solution. Because two points determine a unique straight line, we only need two solutions to draw the whole graph; a third point is taken just as a check. Steps to draw the graph: (i) find at least two (preferably three) solutions and make a table of x and y values, (ii) plot these points on graph paper, (iii) join them with a ruler and extend the line both ways with arrowheads.
6. Equations of lines parallel to the axes
Consider the equation x = a (for example x = 3). In one variable this means a single point on the number line, but in two variables x = 3 means x + 0·y − 3 = 0: x is fixed at 3 while y can be anything. Its graph is a vertical line parallel to the y-axis, passing through (3, 0). Similarly y = b (for example y = 2) keeps y fixed while x varies, giving a horizontal line parallel to the x-axis through (0, 2). Special cases: x = 0 is the y-axis itself, and y = 0 is the x-axis itself.
7. Lines through the origin
When c = 0, the equation ax + by = 0 always has (0, 0) as a solution, because putting x = 0 and y = 0 satisfies it. So its graph is a straight line that passes through the origin. For example, y = 2x passes through (0, 0), (1, 2) and (2, 4). Lines with a non-zero constant term do not pass through the origin.
8. Reading a graph and real-life use
A graph lets us read solutions quickly: pick any x on the line, go up to the line, then read the y value. Linear equations model real situations — for example, converting temperatures (F = (9/5)C + 32), the cost of items at a fixed price, or distance covered at constant speed. Plotting them helps us see how one quantity changes with another in a straight, predictable way.
- Standard form: ax + by + c = 0, a and b not both zero.
- A solution is an ordered pair (x, y) satisfying the equation.
- A linear equation in two variables has infinitely many solutions.
- The graph is always a straight line; two points fix it.
- x = a → vertical line; y = b → horizontal line.
- y = 0 is the x-axis; x = 0 is the y-axis.
- If c = 0, the line passes through the origin (0, 0).
Write 5x = −3y + 2 in standard form ax + by + c = 0 and find its coefficients. Then find two solutions.
- Bring all terms to the left side: 5x + 3y − 2 = 0.
- Compare with ax + by + c = 0 → a = 5, b = 3, c = −2.
- Find a solution: put x = 0 → 3y − 2 = 0 → y = 2/3, so (0, 2/3).
- Find another: put y = 0 → 5x − 2 = 0 → x = 2/5, so (2/5, 0).
Draw the graph of 2x + y = 6 by finding three solutions.
- Rewrite as y = 6 − 2x to make finding y easy.
- Let x = 0 → y = 6 − 0 = 6, giving the point (0, 6).
- Let x = 1 → y = 6 − 2 = 4, giving the point (1, 4).
- Let x = 3 → y = 6 − 6 = 0, giving the point (3, 0).
- Plot (0, 6), (1, 4) and (3, 0) on graph paper and join them with a ruler — all three lie on one straight line.
"Two points, one line." You never need more than two solutions to draw the graph — the third is just your safety check. And remember the order: x first, y second — like reading along the floor before climbing the wall.
The most common mistake is mixing up the order in (x, y) and plotting (3, 0) as if it were (0, 3). Always plot the x-value first (left/right), then the y-value (up/down). Also, when moving a term across the equals sign, change its sign — forgetting this gives the wrong a, b, c.
Q1. Express the equation y = 3x in the form ax + by + c = 0 and write a, b, c.
Answer: Bring all terms to one side: 3x − y = 0, i.e. 3x − y + 0 = 0. Comparing with ax + by + c = 0 we get a = 3, b = −1, c = 0. Since c = 0, this line passes through the origin.
Q2. Find four different solutions of the equation x + 2y = 8.
Answer: Choose values of x and solve for y. (i) x = 0 → 2y = 8 → y = 4 → (0, 4). (ii) x = 2 → 2y = 6 → y = 3 → (2, 3). (iii) x = 4 → 2y = 4 → y = 2 → (4, 2). (iv) x = 8 → 2y = 0 → y = 0 → (8, 0). So four solutions are (0, 4), (2, 3), (4, 2) and (8, 0). Many more are possible.
Q3. If the point (2, 3) lies on the graph of the equation 3x − ky = 0, find the value of k.
Answer: A point on the graph is a solution, so it must satisfy the equation. Put x = 2 and y = 3: 3(2) − k(3) = 0 → 6 − 3k = 0 → 3k = 6 → k = 2. So k = 2.
Q4. The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables to represent this, and give one solution.
Answer: Let the cost of one notebook be Rs x and the cost of one pen be Rs y. "Notebook is twice the pen" means x = 2y, which can be written as x − 2y = 0. One solution: take y = 5 → x = 2(5) = 10, so (x, y) = (10, 5), meaning a notebook costs Rs 10 and a pen costs Rs 5. Since c = 0, the graph passes through the origin.
- ✅ A linear equation in two variables is ax + by + c = 0 with a, b not both zero.
- ✅ A solution is an ordered pair (x, y); there are infinitely many of them.
- ✅ The graph is a straight line and every point on it is a solution.
- ✅ x = a is a vertical line, y = b is a horizontal line, and c = 0 means the line passes through the origin.
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