Introduction to the Equation of a Line

Understanding how to find the equation of a line in standard form is crucial for anyone working with linear equations. In this article, we will explore the process of finding the equation of a line that passes through the point (0, -2) and has a slope of ( frac{2}{3} ). We will also explore alternative methods and provide a comprehensive guide to this concept.

Standard Form of a Line Equation

The standard form of a line equation is given by:

( Ax By C )

where ( A ), ( B ), and ( C ) are constants, and ( A ) is generally a positive integer.

Step-by-Step Method for Finding the Equation of a Line

To find the equation of a line in standard form that passes through the point (0, -2) with a slope of ( frac{2}{3} ), we will follow these steps:

1. Use the Point-Slope Form

The point-slope form of a line equation is given by:

( y - y_1 m(x - x_1) )

where ( m ) is the slope and ( (x_1, y_1) ) is a point on the line.

Given the slope ( m frac{2}{3} ) and the point ( (x_1, y_1) (0, -2) ), we can substitute these values into the point-slope form:

( y - (-2) frac{2}{3}(x - 0) )

This simplifies to:

( y 2 frac{2}{3}x )

2. Convert to Standard Form

To convert the equation ( y 2 frac{2}{3}x ) into standard form, we need to eliminate the fraction and rearrange the terms:

( y 2 frac{2}{3}x )

Multiplying every term by 3 to eliminate the fraction:

( 3y 6 2x )

Subtract ( 2x ) and ( 6 ) from both sides:

( -2x 3y -6 )

To make the coefficient of ( x ) positive, multiply every term by -1:

( 2x - 3y 6 )

This is the equation of the line in standard form.

Alternative Methods for Finding the Equation of a Line

Let's explore another method to find the equation of a line:

General Form of a Line

The general form of a straight line is:

( y mx c )

Given ( m frac{2}{3} ), the equation becomes:

( y frac{2}{3}x c )

Since the line passes through the point (0, -2), we can substitute ( x 0 ) and ( y -2 ) into the equation to find ( c ):

( -2 frac{2}{3}(0) c )

This simplifies to:

( c -2 )

Therefore, the equation of the line is:

( y frac{2}{3}x - 2 )

Using the Formula ( y mx b )

The formula ( y mx b ) where ( m ) is the slope and ( b ) is the y-intercept. Given ( m frac{2}{3} ) and the y-intercept ( b -2 ), the equation is:

( y frac{2}{3}x - 2 )

Using the Point-Slope Formula Again

Starting again with the point-slope form:

( y - y_1 m(x - x_1) )

Substituting ( m frac{2}{3} ) and ( (x_1, y_1) (0, -2) ), we get:

( y - (-2) frac{2}{3}(x - 0) )

This simplifies to:

( y 2 frac{2}{3}x )

Multiplying by 3 to eliminate the fraction:

( 3y 6 2x )

Rearranging to standard form:

( 2x - 3y 6 )

Conclusion

By following these steps, we have successfully determined the equation of a line in standard form that passes through the point (0, -2) with a slope of ( frac{2}{3} ). We have used the point-slope form, the general form, and the slope-intercept form to reach the same conclusion.

The equation of the line in standard form is:

( boxed{2x - 3y 6} )

If you have any more questions or need further assistance with linear equations, feel free to reach out.