Question

The equation of a line with slope 2/3 and passing through (3, - 2) is :

• A. 2x - 6y = 12
• B. 3x - 2y = 12
• C. 2x - 5y = 6
• D. 2x - 3y = 12

Answer 1

Answer:

D. 2x - 3y = 12

Equation of a Line

The general equation of a straight line can be represented in various forms. For this problem, we utilize the point-slope form and the slope-intercept form, and then convert it to the standard form (Ax + By = C).

Point-Slope Form

• The point-slope form of a linear equation is given by: y - y₁ = m(x - x₁)

• Here, m represents the slope of the line, and (x₁, y₁) is a point that the line passes through.

Given Information

• Slope (m) = 2/3

• Point (x₁, y₁) = (3, -2)

Applying the Point-Slope Form

1. Substitute the given values into the point-slope formula:

2. y - (-2) = (2/3)(x - 3)

3. Simplify the equation:

4. y + 2 = (2/3)(x - 3)

Converting to Standard Form (Ax + By = C)

1. To eliminate the fraction, multiply both sides of the equation by 3:

2. 3(y + 2) = 2(x - 3)

3. Distribute the constants:

4. 3y + 6 = 2x - 6

5. Rearrange the terms to match the standard form. Move the 'x' term to the left and the constant term to the right:

6. -2x + 3y = -6 - 6

7. -2x + 3y = -12

8. Multiply the entire equation by -1 to make the coefficient of 'x' positive, which is a common convention for the standard form:

9. 2x - 3y = 12

Alternative Method: Slope-Intercept Form

• The slope-intercept form is y = mx + c, where 'c' is the y-intercept.

• Substitute the slope (m = 2/3):

• y = (2/3)x + c

• Use the given point (3, -2) to solve for 'c':

• -2 = (2/3)(3) + c

• -2 = 2 + c

• c = -2 - 2

• c = -4

• Now, substitute 'c' back into the slope-intercept form:

• y = (2/3)x - 4

• Convert this to the standard form (Ax + By = C) as done previously. Multiply by 3:

• 3y = 2x - 12

• Rearrange:

• -2x + 3y = -12

• Multiply by -1:

• 2x - 3y = 12