Math Problem Statement

Reduce the following line slope interest form and intersect form (a) 2x+3y-6=0

Solution

To reduce the given equation 2x+3y6=02x + 3y - 6 = 0 into slope-intercept form (y=mx+by = mx + b) and standard intercept form (x/a+y/b=1x/a + y/b = 1), follow these steps:


1. Slope-Intercept Form (y=mx+by = mx + b):

The general slope-intercept form of a line is: y=mx+by = mx + b

Start with the given equation: 2x+3y6=02x + 3y - 6 = 0

Rearrange to isolate yy: 3y=2x+63y = -2x + 6

Divide through by 3: y=23x+2y = -\frac{2}{3}x + 2

Here:

  • Slope (mm) = 23-\frac{2}{3}
  • Y-intercept (bb) = 2

2. Intercept Form (xa+yb=1\frac{x}{a} + \frac{y}{b} = 1):

The general intercept form of a line is: xa+yb=1\frac{x}{a} + \frac{y}{b} = 1

Start with the given equation: 2x+3y6=02x + 3y - 6 = 0

Solve for the intercept form: 3y=2x+63y = -2x + 6

Divide through by 6 to normalize: 2x6+3y6=1\frac{-2x}{6} + \frac{3y}{6} = 1

Simplify the fractions: x3+y2=1\frac{x}{-3} + \frac{y}{2} = 1

Here:

  • X-intercept (aa) = 3-3
  • Y-intercept (bb) = 22

Final Results:

  1. Slope-Intercept Form: y=23x+2y = -\frac{2}{3}x + 2

  2. Intercept Form: x3+y2=1\frac{x}{-3} + \frac{y}{2} = 1

Would you like a detailed plot or further clarifications?


Expanding Questions:

  1. How can the slope be interpreted geometrically in the graph of a line?
  2. What happens to the intercepts if the equation coefficients change?
  3. How would you convert intercept form back to slope-intercept form?
  4. What is the geometric significance of having one intercept at zero?
  5. How do parallel and perpendicular lines relate to the slope of this line?

Tip: Always ensure the line equation is simplified before extracting slope or intercepts to avoid calculation errors.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Slope-Intercept Form
Intercept Form

Formulas

Slope-Intercept Form: y = mx + b
Intercept Form: x/a + y/b = 1

Theorems

Properties of Linear Equations

Suitable Grade Level

Grades 8-10