Verify whether the given equation can represent a linear function. Explain your reasoning.
(a)
$\frac{3}{4}$x +
$\frac{1}{2}$y = 1 (b) x  xy + y = 3
Solution:
The equation (a) is a first degree equation in x and y and hence it represents a linear function. The equation can be rewritten as
$\frac{1}{2}$y = 
$\frac{3}{4}$x + 1 Separate variables
2 .
$\frac{1}{2}$y = (
$\frac{3}{4}$x + 1) . 2 (Multiplying by 2)
y =
$\frac{3}{2}$x + 2
f(x) =
$\frac{3}{2}$x + 2 . Linear function of the form f(x) = mx + b.
The equation (b) contains a second degree term xy. Thus the equation cannot be written in the form y = mx + b and hence does not represent a linear function.
Example: Check whether the data given in the table represents a linear function. If yes, find the function rule that defines the table.
x

f(x)
 Difference in f(x) 
4 
73  80 
8 
153  80 
12 
233  80 
16 
313  80 
20 
393  80 
24 
473  80 
It can be noted that for a constant change of 4 in x values, f(x) increases by a constant value of 80. Hence the table represents a linear function.
m = Rate of change =
$\frac{Change\ in\ f(x)}{Change\ in\ x}$ =
$\frac{80}{4}$ = 20
Hence f(x) is of the form, f(x) = 20x + b.
Here (4, 73) is an ordered pair that satisfies the equation.
=> f(4) = 20(4) + b = 73 ⇒ b = 73  80 = 7.
Hence the rule that defines the table is f(x) = 20x  7.