y=mx+b

y = mx + b is the slope-intercept form of a staight line. In the equation y = mx + b for a straight line, m is called the slope of the line and b is the y-intercept of a line. y = mx+b, where 

y ⇒  how far up or down is the line, 

x  ⇒ how far along is the line,   

b ⇒ the value of y when x = 0 and

m ⇒ how steep the line is.

This is determined by m = (difference in y coordinates)/ (difference in x coordinates). Note that difference in y coordinates is indicated as rise or fall and difference in x coordinates is indicated as run.

y = mx + b is the formula used to find the equation of a straight line, when we know the slope(m) and the y-intercept (b) of the line. To determine m, we apply a formula based on the calculations. Let's derive this formula using the equation for the slope of a line. Let us consider a line whose slope is 'm' and whose y-intercept is 'b'. Let (x,y) be any other random point on the line whose coordinates are not known. We obtain the graph as follows.

We know that the equation for the slope of a line in the slope-intercept form is y = mx+b

Rewriting this, we get m = (y-b) / x 

Thus the formula to find m = change in y/ change in x

Let us derive the formula to find the value of the slope if two points \((x_{1},y_{1})\) and \((x_{2},y_{2})\) on the straight line are known. Then we have \(y_{1} = mx_{1} + b\) and \(y_{2} = mx_{2} + b\)

We know that, slope = change in y/ change in x

Substituting the values of y₁ and y₂, we get \[\begin{align}\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}&= \dfrac{(mx_{2}+b) - (mx_{1}+b)}{x_{2}-x_{1}}\\\\&=\dfrac{mx_{2}-mx_{1}}{x_{2}-x_{1}}\\\\&= \dfrac{m(x_{2}-x_{1})}{x_{2}-x_{1}}\\\\ &=m\end{align}\]

Thus we find that the slope (m) is calculated as (change in y)/ (change in x)

m = (difference in y coordinates)/ (difference in x coordinates)

To find the y-intercept or 'b', substitute the value of 'x' as 0 in the equation of a straight line, which is of the form Ax + By + C = 0. Consider an equation of a straight line : 3x + 5y = 10. To find the y-intercept, substitute the value of 'x' as 0 in the equation and solve for 'y'. On substituting 'x = 0' in the equation 3x + 5y =10, we get, 3(0) + 5y = 10
⇒5y = 10 and thus y = 10/5 ⇒ y = 2 or 'b' = 2.

If the slope 'm' and y-intercept 'b' are given, then the equation of the straight line can be written in the form of 'y = mx +b'. For example, if the slope(m) for a line is 2 and the y-intercept 'b' is -1, then the equation of the straight line is written as y = 2x - 1.  The slope value can be positive or negative. As we discussed in the earlier sections, in y = mx + b, 'm' represents the slope of the equation. To find the slope of a line, given its equation, we have to rearrange its terms to the slope-intercept form y = mx + b. Here, 'm' gives the slope and 'b'  gives the y-intercept of the equation.

Let us consider the equation 2x + 3y = 6. We are required to find the slope and the y-intercept from the equation which is of the form Ax + By = C

We rewrite the standard form of the equation of the line to the slope-intercept form y = mx + b. 

2x + 3y = 6
3y = 2x + 6
y = (-2/3) x + 2

Comparing the final equation with y = mx + b, we obtain the slope of the equation is m = -2/3 and the y-intercept of the equation is, b = 2 or (0,2).

Important Notes:

• The equation of the slope-intercept form of a line whose slope is 'm' and whose y-intercept is 'b' or (0,b) is y = mx + b.
• The equation of a horizontal line passing through (a,b) is of the form y = b.
• The equation of a vertical line passing through (a,b) is of the form x = a.
• m is calculated using the formula rise over run or (change in y)/ (change in x)

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