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Consider the set of all ordered pairs of real numbers (x, y) which satisfy the condition that the sum of the first and second components is +5.
    x + y = +5

Some ordered pairs that belong to this function are:
    (+5, 0), (3 1/2, 1 1/2), (+6, -1)

This set of ordered pairs may also be described as the set of all ordered pairs of real numbers (x, y) such that
    y = -1 - x + +5

This set of ordered pairs of real numbers is a function whose domain and range is the set of all real numbers. The graph of this function is a line:

Algebra

The points on the graph that correspond to the ordered pairs (+2, +3) and (+7, -2) are marked on the graph. You may find it worthwhile to locate the points on the graph that correspond to the following ordered pairs:
    (0, +5), (+1.5, +3.5), (-1, +6)

Since the graph of the function is a line, the function is called a linear function, and the equation
    y = -1 - x + +5
is called a linear equation.

Since the graph of a linear function is a straight line, many geometric problems that involve lines and line segments may be solved by algebraic methods. The properties of linear functions are studied in analytic geometry and calculus.

Another example of a linear function is the set of all ordered pairs of real numbers (x, y) such that

y = x + +1

Some ordered pairs that belong to this function are:

(0, +1), (-1, 0), (+2.5, +3.5), (-2.5, -1.5)


The graph of this function is a line:

Here are other examples of linear equations:

y = x + -6 y = +5x + -2 y = -4x + +1

In fact, if 'a' and 'b' are real numbers, and 'a' <> 0,

y = ax + b

is a linear equation. The corresponding function is a linear function, and its graph is a straight line.



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