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**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 (x

Hence,

Ax

Ax

Ax

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 = (y

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

**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

\(\therefore\) (A

**References**

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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