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 $\frac{7}{2}$.

● The Straight Line

• Straight Line
• Slope of a Straight Line
• Slope of a Line through Two Given Points
• Collinearity of Three Points
• Equation of a Line Parallel to x-axis
• Equation of a Line Parallel to y-axis
• Slope-intercept Form
• Point-slope Form
• Straight line in Two-point Form
• Straight Line in Intercept Form
• Straight Line in Normal Form
• General Form into Slope-intercept Form
• General Form into Intercept Form
• General Form into Normal Form
  
• Point of Intersection of Two Lines
  
• Concurrency of Three Lines
• Angle between Two Straight Lines
• Condition of Parallelism of Lines
• Equation of a Line Parallel to a Line
• Condition of Perpendicularity of Two Lines
• Equation of a Line Perpendicular to a Line
• Identical Straight Lines
• Position of a Point Relative to a Line
• Distance of a Point from a Straight Line
• Equations of the Bisectors of the Angles between Two Straight Lines
• Bisector of the Angle which Contains the Origin
• Straight Line Formulae
• Problems on Straight Lines
• Word Problems on Straight Lines
• Problems on Slope and Intercept

11 and 12 Grade Math 

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