Example Use the distance formula to determine if three points A(5, 3), B(1, 1) and C(4, 3) are collinear.
Solution : In D ABC, d (A,C) £ d (A,B) + d (B, C). . . (by the triangle inequality)
and d (A, C) = d (A, B) + d (B, C) if and only if A, B and C are colinear points. Thus,
Since
\ The three points A, B and C are collinear.
Example Graph the equation 2x + y = 7 \ y = 2x + 7
Solution : When x = 1, y = 2 (1) + 7 i.e. y = 9
When x = 0, y = 2 (0) + 7 i.e. y =7
When x = 1, y = 2 (1) + 7 i.e. y = 5
Observation : The solution of the equation 2x + y = 7, when plotted, forms a straight line.
Conclusion : The equation whose graph is in a straight line is called a linear equation.
Thus every first degree equation in x and y of the form ax + by + c = 0 where a, b and c are real numbers, always represents a straight line.
There are equations in which a variable is raised to power, shows division by variable, involves variables with square roots, or variables multiplied together. Such equations will not from a straight line. When the solutions are graphed then such equations are called nonlinear equations.
Example Graph the equation y = x^{2} 2
Solution : When
i.e.
From this example, it is clear that solutions of nonlinear equations, when plotted give a curved line.
