From one variable to two
In Class 8 you solved equations like 2x + 5 = 11, which have exactly one solution. Many real situations involve two unknowns.
Example. A pen costs ₹x and a notebook costs ₹y. Four pens and three notebooks cost ₹100 in all. This gives 4x + 3y = 100.
An equation that can be written as
ax + by + c = 0, where a, b and c are real numbers and a and b are not both zero,
is a linear equation in two variables.
Equation | a | b | c |
|---|---|---|---|
2x + 3y = 12, i.e. 2x + 3y − 12 = 0 | 2 | 3 | −12 |
x = 4y, i.e. x − 4y + 0 = 0 | 1 | −4 | 0 |
y = 5, i.e. 0·x + 1·y − 5 = 0 | 0 | 1 | −5 |
x = −2, i.e. 1·x + 0·y + 2 = 0 | 1 | 0 | 2 |
The last two show that equations in one variable can also be treated as equations in two variables.
Solutions
A solution is a pair of values (x, y) that makes the equation true.
For x + 2y = 6:
x = 0 gives 2y = 6, so y = 3. The pair (0, 3) is a solution.
y = 0 gives x = 6, so (6, 0) is a solution.
x = 2 gives 2y = 4, so (2, 2) is a solution.
For each value of x you choose, you get a matching value of y. So a linear equation in two variables has infinitely many solutions.
To check whether a pair is a solution, substitute it. Is (4, 1) a solution of x + 2y = 6? 4 + 2 = 6. Yes. Is (1, 4)? 1 + 8 = 9 ≠ 6. No. Order matters: (x, y) always lists x first.
Drawing the graph
Find at least two solutions (a third is a good check). Taking x = 0 and y = 0 is often easiest.
Plot them on the Cartesian plane.
Join them with a straight line and extend it in both directions.
Example. Draw the graph of 2x + y = 4.
x | 0 | 2 | 1 |
|---|---|---|---|
y | 4 | 0 | 2 |
Plot (0, 4), (2, 0) and (1, 2). They lie on one straight line. This is why such equations are called linear.
Two key facts:
Every point on the line is a solution of the equation.
Every solution of the equation is a point on the line.
Through any one point, infinitely many lines pass, so infinitely many linear equations in two variables have that point as a solution. For example, (1, 2) satisfies 2x + y = 4, y = 2x, x + y = 3 and many more.
Special lines
Equation | Graph |
|---|---|
y = 0 | the x-axis |
x = 0 | the y-axis |
y = k (k ≠ 0) | a line parallel to the x-axis, k units from it |
x = k (k ≠ 0) | a line parallel to the y-axis, k units from it |
y = mx | a line through the origin |
Example. The graph of x = 3 is a vertical line through (3, 0). Every point on it has x-coordinate 3, whatever its y-coordinate.
Using a graph to answer questions
Example. A taxi charges ₹30 for the first kilometre and ₹15 for every kilometre after that. If the distance is x km and the fare is ₹y, then y = 30 + 15(x − 1), which is y = 15x + 15. From its graph, or by substituting, a 10 km ride costs 15 × 10 + 15 = ₹165.
Summary
ax + by + c = 0 (a, b not both 0) is a linear equation in two variables.
It has infinitely many solutions, each an ordered pair (x, y).
Its graph is a straight line; each point on it is a solution.
y = k is parallel to the x-axis and x = k is parallel to the y-axis.