# General Form into Slope-intercept Form

We will learn the transformation of general form into slope-intercept form.

To reduce the general equation Ax + By + C = 0 into slope-intercept form                (y = mx + b):

We have the general equation Ax + By + C = 0.

If b ≠ 0, then from the given equation we get,

By = - Ax - C (Subtracting ax from both sides)

⇒ y= - A/Bx - C/B, [Dividing both sides by b (≠0).

⇒ y = (-$$\frac{A}{B}$$)x + (-$$\frac{C}{B}$$)

Which is the required slope-intercept form (y = mx + b) of the general form of line Ax + By + C = 0, where m = -$$\frac{A}{B}$$, b = -$$\frac{C}{B}$$

Thus, for the straight line Ax + By + C = 0,

m = slope = -$$\frac{A}{B}$$ = - $$\frac{\textrm{Coefficient of x}}{\textrm{Coefficient of y}}$$

Note:

To determine the slope of a line by the formula m = - $$\frac{\textrm{Coefficient of x}}{\textrm{Coefficient of y}}$$ first transfer all terms in the equation on one side.

Solved examples on transformation of general equation into slope-intercept form:

1. Transform the equation of the straight line 2x + 3y - 9 = 0 to slope intercept form and find its slope and y-intercept.

Solution:

The given equation of the straight line 2x + 3y - 9 = 0

First subtract 2x from both sides.

⇒ 3y - 9 = -2x

Now add 9 on both sides

⇒ 3y = -2x + 9

Then divide both sides by 3

⇒ y = (-$$\frac{2}{3}$$)x + 3, which is the required slope-intercept form of the given straight line 2x + 3y - 9 = 0.

Therefore, slope of the given line (m) = -$$\frac{2}{3}$$ and y-intercept = 3.

2. Reduce the equation -5x + 2y = 7 into slope intercept form and find its slope and y-intercept.

Solution:

The given equation of the straight line -5x + 2y = 7.

Now solve for y in terms of x.

⇒ 2y = 5x + 7

⇒ y = ($$\frac{5}{2}$$)x + $$\frac{7}{2}$$, which is the required slope-intercept form of the given straight -5x + 2y = 7.

Therefore, slope of the given straight line $$\frac{5}{2}$$ and y-intercept $$\frac{7}{2}$$.

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