# Linear Function

**Linear function** is given as `y=f(x)=mx+b`.

`m` is called **slope** and `b` is called **y-intercept**. Graph of the linear function is **line**. Since there are two parameters in the linear function (m and b) it is enough two points to uniquely identify line.

If line passes through 2 points `(x_1,y_1)` and `(x_2,y_2)` then `color(blue)(m=(y_2-y_1)/(x_2-x_1))`.

`m` shows rate of change of function. For example, for function `y=3x+2` if we increase `x` by 1 unit, then `y` will increase by 3 units. A characteristic feature of linear functions is that they grow at a constant rate.

Sometimes linear function can be given in the form `ax+cy=d`, where `c!=0`. We can easily convert it into standard form by dividing both sides of equation by `c`: `y=-a/c x+d/c`. Here `m=-a/c` and `b=d/c`.

**Example 1.** Find equation of line that passes through points `(2,-3)` and `(-5,1)`.

Slope of the line is `m=(1-(-3))/(-5-2)=-4/7`.

Now, equation of line can be rewritten as `y=-4/7 x+b`. Since line passes through point `(2,-3)` then `-3= -4/7 *2+b`. From this `b=-13/7`.

Thus, equation of line is `y=-4/7 x-13/7`.

**Example 2**. Draw line whose equation is `3x+2y=6`.

We need to points to draw a graph.

If `x=0` then `3*0+2y=6` or `y=3`. Therefore, first point is `(0,3)`.

If `y=0` then `3x+2*0=6` or `x=2`. Therefore, second point is `(2,0)`.

Now draw two found points and line through them.

Graph is shown on figure.

Suppose that we have two lines `y=m_1x+b_1` and `y=m_2x+b_2`.

Lines are **parallel** if `m_1=m_2`. If in addition `b_1=b_2` then lines are same.

Lines are **perpendicular** if `m_1m_2=-1`.

**Example 3**. Find equation of line that is perpendicular to line `4x+3y=2` and passes through point `(-1,3)`.

Let's rewrite given line in standard form: `y=-4/3 x+2/3`.

If slope of required line is `m`, then since lines are perpendicular we have that `m*(-4/3)=-1` or `m=3/4`.

Thus, equation of line is `y=3/4 x+b`. To find constant `b` we use the fact that line passes through point `(-1,3)`: `3=3/4 *(-1)+b` or `b=15/4`.

So, equation of line is `y=3/4x+15/4` or `4y-3x=15`.