Introduction to the Problem

Determining the equation of a line that is parallel to a given line and passes through a specific point is a fundamental concept in algebra and geometry. This guide walks you through the process step-by-step, ensuring that you understand the underlying principles and can apply the method to similar problems.

Step 1: Converting to Slope-Intercept Form

To solve this problem, our first step is to convert the given line equation 2x 3y 6 into its slope-intercept form, y mx b, where m represents the slope and b is the y-intercept.

Equation: 2x 3y 6

Start by isolating y:

Step 1.1: Subtract 2x from both sides

3y -2x 6

Step 1.2: Divide by 3

y -frac{2}{3}x 2

From this equation, we can see that the slope (m) of the line is -frac{2}{3}.

Step 2: Using the Slope of a Parallel Line

Parallel lines have the same slope. Therefore, the line we need to find will also have a slope of -frac{2}{3}.

Step 3: Finding the Equation Using Point-Slope Form

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

Point-Slope Form: y - y_1 m(x - x_1)

Here, (x_1, y_1) is the given point (-2, 3) and m is the slope -frac{2}{3}.

Plugging in the Values

Substitute the point and the slope into the point-slope form:

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

Distribute the slope on the right-hand side:

Equation: y - 3 -frac{2}{3}x - frac{4}{3}

Add 3 to both sides to get the equation in slope-intercept form:

Equation: y -frac{2}{3}x - frac{4}{3} frac{9}{3}

Simplify the right-hand side:

Equation: y -frac{2}{3}x frac{5}{3}

Final Equation

The equation of the line that passes through the point (-2, 3) and is parallel to the line 2x 3y 6 is:

Final Equation: y -frac{2}{3}x frac{5}{3}

Conclusion

Understanding the process of finding the equation of a line parallel to a given line and passing through a specific point is crucial in both algebra and geometry. By mastering these techniques, you can solve a wide range of problems and apply the concept to more complex scenarios.