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 \(\frac{7}{2}\).

11 and 12 Grade Math 

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