How to write the equation of a line parallel to another given a point

📚 Understanding Parallel Lines

In mathematics, parallel lines are lines in a plane that never intersect or touch each other. More formally, two lines in a plane are said to be parallel if they have the same slope. Understanding this concept is crucial in coordinate geometry. Let's dive in!

📜 A Brief History

The concept of parallel lines dates back to ancient Greece. Euclid's postulates in 'The Elements' laid the foundation for understanding parallel lines and their properties. The parallel postulate, in particular, has been a subject of intense study and debate throughout mathematical history.

📐 Key Principles for Writing Parallel Line Equations

• 🔍 Identifying the Slope: The most crucial step is to identify the slope of the given line. If the equation is in slope-intercept form ($y = mx + b$), the slope is simply $m$.
• 🤝 Parallel Lines Share the Same Slope: Parallel lines have equal slopes. Use the slope from the given line.
• 📍 Using the Point-Slope Form: Given a point $(x_1, y_1)$ and the slope $m$, the point-slope form of a line is $y - y_1 = m(x - x_1)$.
• ✍️ Converting to Slope-Intercept Form (Optional): You can rewrite the equation in slope-intercept form ($y = mx + b$) for clarity or to match a specific required format.

✏️ Step-by-Step Guide

1. Identify the slope of the given line. Let's say the given line is $y = 2x + 3$. The slope is $2$.
2. Use the same slope for the parallel line. So, the slope of the parallel line is also $2$.
3. Use the given point. Let's say the point is $(1, 5)$.
4. Plug the slope and point into the point-slope form: $y - 5 = 2(x - 1)$.
5. Simplify the equation: $y - 5 = 2x - 2$.
6. Convert to slope-intercept form: $y = 2x + 3$.

🧮 Real-World Examples

Example 1:

Write the equation of a line parallel to $y = 3x - 2$ that passes through the point $(2, 7)$.

• 📝 Slope of the given line: $m = 3$
• 📍 Point: $(2, 7)$
• ✍️ Point-slope form: $y - 7 = 3(x - 2)$
• 💡 Simplified equation: $y = 3x + 1$

Example 2:

Write the equation of a line parallel to $y = -x + 5$ that passes through the point $(-1, 4)$.

• 📐 Slope of the given line: $m = -1$
• 📌 Point: $(-1, 4)$
• 🧪 Point-slope form: $y - 4 = -1(x + 1)$
• ➗ Simplified equation: $y = -x + 3$

📝 Practice Quiz

Here are some practice problems to test your understanding:

1. Find the equation of a line parallel to $y = 4x - 1$ passing through $(0, 2)$.
2. Find the equation of a line parallel to $y = -2x + 3$ passing through $(1, -1)$.
3. Find the equation of a line parallel to $y = \frac{1}{2}x + 4$ passing through $(2, 5)$.
4. Find the equation of a line parallel to $y = -\frac{3}{4}x - 2$ passing through $(-4, 1)$.
5. Find the equation of a line parallel to $2x + y = 5$ passing through $(3, 0)$.
6. Find the equation of a line parallel to $x - 3y = 6$ passing through $(-2, 2)$.
7. Find the equation of a line parallel to $5x + 2y = 8$ passing through $(-1, -3)$.

Answers:

1. $y = 4x + 2$
2. $y = -2x + 1$
3. $y = \frac{1}{2}x + 4$
4. $y = -\frac{3}{4}x - 2$
5. $y = -2x + 6$
6. $y = \frac{1}{3}x + \frac{8}{3}$
7. $y = -\frac{5}{2}x - \frac{11}{2}$

💡 Conclusion

Writing the equation of a line parallel to another is a fundamental skill in algebra and geometry. By understanding the principles of slope and using the point-slope form, you can easily find the equation of any parallel line. Practice makes perfect, so keep working through examples! 🎉