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Linear Functions

Linear functions
Linear functions are the simple function of all the types of functions. A linear function takes a number x as input and returns the number mx+b as output, b and m are constants. In words, x gets multiplied by m (this is called a scaling by factor m) and then gets b added on (this is called the shift by an amount b). Using function notation the linear function looks like this:
f(x)= mx+b.
If we let y = f(x) then looks like this:
y = mx+b.
This is called the equation of a straight line.
A typical use of the linear function is to convert from one set of units to another. A simple example is if i is a distance measured in inches and for c is the same distance measured in centimeters; then c = 2.54i. This is just a scaling. A more complicated example is if that c is a temperature measured in celsius degrees and f is the same temperature measured in fahrenheit degrees; then f = 1.8c + 32. This is a scaling and a shift.

Linear Equation Ax + By + C = 0.
The general equation of first degree in x & y is Ax + By + C = 0, where A, B, C are constants and A and B are not simultaneously zero. This is also called the linear equation in x and y as we shall subsequently see that it always represents a straight line.
To prove that Ax + By + C = 0 always represents a straight line. Let P (x1, y1), Q (x2, y2), R (x3, y3) be any three points on the locus represented by the equation Ax + By + C = 0. The coordinates of the points must satisfy the equation.
Hence,
Ax1 + By1 + C = 0
Ax2 + By2 + C = 0
Ax3 + By3 + C = 0.
From the first equations , by the rule of cross multiplication, we have
\(\frac{A}{y_1 – y_2}\) = \(\frac{B}{x_2 – x_1}\) = \(\frac {OL}{x_1y_2 – x_2y_1}\) = k, say
A = (y1  – y2) k, B = (x2  –  x1) k, C = (x1y2  – x2y1) k

Or,

The L.H.S. is twice the area of the triangle PQR. Hence the area of the triangle is zero, which means that P, Q, R lie in a straight line.
Thus the linear equation represents a straight line.
The converse is easily seen to be true. Because equation of any line not parallel to y-axis can be written as y = mx + c, and that of any line parallel to y-axis is x = a.
Both of these equations are linear.

Reduction of the Linear Equation to Three Standard Forms
The linear equation Ax + By + C = 0 can be reduced to the three standard forms : slop-intercept form, double intercept form and the normal form.

Reduction to the Slope Intercept Form
Ax + By + C = 0 can be written as By = - Ax – C
i.e. y = - \(\frac{A}{B}\)x - \(\frac {C}{B}\) which is in the form y = mx + c
where slope (m) = - \(\frac {A}{B}\) and y – intercept ( c ) = - \(\frac {C}{B}\).

Reduction to the Double Intercept Form
Ax + By + C = 0 can be written as
Ax + By = - C
or, \(\frac {A}{-C}\)x + \(\frac {B}{-C}\)y = 1
or \(\frac {x}{-\frac{C}{A}}\) + \(\frac {y}{-\frac{C}{B}}\) = 1, which is of thee form \(\frac {x}{a}\)+\(\frac {y}{b}\) = where x intercept = a = - \(\frac {C}{A}\), y intercept = b -\(\frac {C}{B}\),

Reduction to the Normal Form
The equations,
Ax +By + C = 0
And x cos \(\alpha\) + y sin \(\alpha\) - p = 0
Will represent one and the same straight line if their corresponding coefficients are proportional.
i.e. cos \(\alpha\)
\(\frac{cos \alpha}{A}\)=\(\frac{sin \alpha}{B}\)= -\(\frac{P}{C}\) = K, say
so that cos \(\alpha\) = Ak, sin \(\alpha\) = Bk. - p = Ck
\(\therefore\) (A2)(k2) + (B2(K2)) = 1

References
Bajracharya, D.R. Basic Mathematics. Kathmandu: Sukunda Pustak Bhawan, 2068.
Izrail Moiseevich Gelfand (1961), Lectures on Linear Algebra, Interscience Publishers, Inc., New York. Reprinted by Dover, 1989.
James Stewart (2012), Calculus: Early Transcendentals, edition 7E, Brooks/Cole


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