Introduction to Parallel Lines and Line Equations

Understanding the concept of parallel lines and their equations is essential in various fields of mathematics, including geometry and calculus. One common task is finding the equation of a line that is parallel to a given line and passes through a specific point. This article will walk you through the steps to achieve that, using the example of the line 3xy 3 and a point it should pass through, namely -4, -5.

Step-by-Step Guide to Solving the Problem

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

The given equation is 3xy 3. To find the slope and then the equation of a parallel line, we first need to convert this equation into the slope-intercept form y mx b.

Starting with the equation:
3xy 3

Divide every term by 3 to simplify:

xy 1

Rearrange to isolate y on one side:

y -3x 3

The equation is now in the form y mx b, where m (the slope) is -3, and b (the y-intercept) is 3.

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

Parallel lines have the same slope. Therefore, the slope of the line we are looking for is also -3. The general form of the equation with this slope is:

y -3x b

We need to find the value of b (the y-intercept) such that the line passes through the point -4, -5. To do this, we substitute x -4 and y -5 into the equation:

-5 -3(-4) b

Simplify the right-hand side:

-5 12 b

Isolate b by subtracting 12 from both sides:

b -17

Thus, the equation of the line is:

y -3x - 17

Step 3: Verify the Solution

To verify our solution, we can substitute the point -4, -5 back into the equation:

-5 -3(-4) - 17

This simplifies to:

-5 12 - 17

-5 -5

Since this is a true statement, our solution is verified.

Conclusion

In this article, we have demonstrated how to find the equation of a line that is parallel to the line 3xy 3 and passes through the point -4, -5. The final equation is y -3x - 17. This process involves understanding the concept of slope, converting the given equation to the slope-intercept form, and using the point-slope form to find the y-intercept of the new line.

Additional Resources

For further study and practice, you may want to explore additional resources such as:

The official Google Search Console documentation for advanced SEO techniques. Interactive online platforms that offer practice problems in algebra and geometry. Textbooks and online courses on linear equations and coordinate geometry.