What is the Equation of a Line Parallel to 2x 3y - 1 0 and Passing Through a Given Point?

To find the equation of a straight line that is parallel to the given line 2x 3y - 1 0 and passes through the point (-1, 2), we need to follow a series of steps. This involves understanding the slope of parallel lines and using the point-slope form of the line equation.

Step 1: Convert the Given Line to Slope-Intercept Form

The first step is to convert the equation 2x 3y - 1 0 into the slope-intercept form, y mx b, where m represents the slope of the line.

Given Equation: 2x 3y - 1 0

Solve for y:

3y -2x 1

Divide by 3:

y -(frac{2}{3})x (frac{1}{3})

Thus, the slope (m) of the given line is (-frac{2}{3}).

Step 2: Use the Slope and Point to Find the Equation of the Parallel Line

Since parallel lines have the same slope, the line we are looking for will also have a slope of (-frac{2}{3}). We can use the point-slope form of the line equation, which is:

y - y1 m(x - x1)

Here, (x1, y1) (-1, 2) and m (-frac{2}{3}).

Substitute the values:

y - 2 -(frac{2}{3})(x - (-1))

Simplify:

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

Bring 2 to the right side:

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

Convert 2 to a fraction to combine constants:

y -(frac{2}{3})x (frac{2}{3}) (frac{6}{3})

Combine the constants:

y -(frac{2}{3})x (frac{8}{3})

Convert back to standard form:

3y -2x 8

Multiply through by 3 to eliminate the fractions:

2x 3y - 4 0

Thus, the equation of the line parallel to 2x 3y - 1 0 and passing through the point (-1, 2) is:

Final equation: 2x 3y - 4 0

Alternative Method: Swapping Coefficients and Negating One Term

Another approach involves directly swapping and negating the coefficients of the original line's equation. Let's start with the original equation:

2x 3y - 1 0

Swapping the coefficients of x and y, we get 3x 2y. Negating one term, specifically the constant term, we get:

3x - 2y - 3 - 1(-1 - 2) 0

Simplifying:

3x - 2y 3 2 0

Combining constants:

3x - 2y 5 0

Thus, the line parallel to 2x 3y - 1 0 and passing through the point (-1, 2) is:

3x - 2y 5 0

Note: In both methods, the constant term and the coefficients are adjusted to align with the given point and parallel line condition.