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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 co-linear 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 = x2 -2

Solution : When

i.e.

From this example, it is clear that solutions of nonlinear equations, when plotted give a curved line.

Index

8.1 Definition
8.2 Relations, Graphs and Symmetry
8.3 Slopes and Intercepts

Chapter 9

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