Through 4 2 Parallel To Y 3 4x 5

How to Find the Equation of a Line Through (4, 2) Parallel to y = (3/4)x + 5

Understanding how to derive the equation of a line that passes through a specific point and runs parallel to another line is a foundational skill in coordinate geometry. This process relies on the properties of parallel lines and the algebraic tools used to describe linear relationships. In this article, we will break down the steps to solve the problem: Find the equation of the line passing through the point (4, 2) and parallel to the line y = (3/4)x + 5. By the end, you’ll not only have the solution but also a deeper understanding of why these steps work.

Step 1: Identify the Slope of the Given Line

The first step in solving this problem is recognizing that parallel lines share the same slope. The given line, y = (3/4)x + 5, is already in slope-intercept form (y = mx + b), where m represents the slope. Here, the slope (m) is 3/4. Since parallel lines never intersect and maintain a constant distance between them, the line we are trying to find must also have a slope of 3/4.

Step 2: Use the Point-Slope Form to Incorporate the Given Point

Now that we know the slope of our desired line is 3/4, we need to ensure it passes through the point (4, 2). To do this, we use the point-slope form of a line’s equation:
$ y - y_1 = m(x - x_1) $
Here, (x₁, y₁) is the point the line passes through, and m is the slope. Substituting the known values:
$ y - 2 = \frac{3}{4}(x - 4) $
This equation now represents a line with slope 3/4 that passes through (4, 2).

Step 3: Simplify to Slope-Intercept Form

To express the equation in the more familiar slope-intercept form (y = mx + b), we simplify the point-slope equation:

1. Distribute the slope on the right-hand side:
   $ y - 2 = \frac{3}{4}x - 3 $
2. Add 2 to both sides to isolate y:
   $ y = \frac{3}{4}x - 3 + 2 $
3. Combine like terms:
   $ y = \frac{3}{4}x - 1 $
   Thus, the equation of the line is y = (3/4)x - 1.

Scientific Explanation: Why Parallel Lines Share the Same Slope

Parallel lines are defined as lines in a plane that never meet, no matter how far they are extended. This geometric property is mathematically enforced by their slopes. If two lines have different slopes, they will eventually intersect. Conversely, if two lines have the same slope, they will remain equidistant and never converge.

The slope of a line measures its steepness and direction. For example, a slope of 3/4 means the line rises 3 units vertically for every 4 units it moves horizontally. A parallel line must mimic this behavior to maintain the same orientation, hence the identical slope.

FAQ: Common Questions About Parallel Lines

**Q: Why can’t two parallel

lines have different slopes?**

A: By definition, parallel lines are equidistant and never intersect. If two lines had different slopes, they would eventually cross, contradicting the definition of parallelism. The identical slope ensures they maintain a constant direction and distance from each other.

Q: Can a horizontal line be parallel to a vertical line?

A: No, a horizontal line has a slope of 0, while a vertical line has an undefined slope (infinity). These lines are perpendicular, not parallel, as they intersect at a 90-degree angle.

Q: How do I find the equation of a line parallel to a given line but passing through a different point?

A: Use the slope of the given line and apply the point-slope form with the new point. For example, if the given line is y = 2x + 3 and you want a parallel line through point (1, 5), use the slope 2 and the point-slope form: y - 5 = 2(x - 1), which simplifies to y = 2x + 3.

Conclusion

By following these steps, we have successfully determined the equation of a line that is parallel to y = (3/4)x + 5 and passes through the point (4, 2). The resulting equation, y = (3/4)x - 1, encapsulates the fundamental properties of parallel lines, emphasizing the crucial role of slope in defining their behavior. Understanding these concepts not only equips us to solve similar problems but also deepens our appreciation for the geometric and algebraic principles that govern parallel lines.

Beyond the algebraic manipulation, visualizing parallel lines on a coordinate plane reinforces why slope consistency is essential. When you plot the original line y = (3/4)x + 5 and the parallel line y = (3/4)x − 1, you’ll notice that every vertical slice (i.e., for any fixed x) yields y values that differ by exactly 6 units—the constant b difference. This uniform vertical offset guarantees that the lines never converge, no matter how far you extend them in either direction.

Practical Applications

1. Engineering Drafting – In technical drawings, parallel lines represent edges of objects that must stay a fixed distance apart (e.g., the opposite sides of a rectangle). Maintaining identical slopes ensures that manufactured parts fit together correctly.
2. Road Design – Lanes on a highway are modeled as parallel lines. Engineers use slope calculations to guarantee that lanes remain evenly spaced across curves and gradients, enhancing safety.
3. Computer Graphics – Rendering algorithms rely on parallelism to create textures and patterns. When shading a surface, a set of parallel scan‑lines with the same slope ensures uniform color distribution across the image.

Quick Practice Problems

1. Find the equation of a line parallel to y = −2x + 7 that passes through (−3, 4).
   Solution: Use slope −2 → y − 4 = −2(x + 3) → y = −2x − 2.
2. Determine whether the lines 3x − 4y = 12 and 6x − 8y = 20 are parallel.
   Solution: Rewrite each in slope‑intercept form; both yield slope 3/4, so they are parallel (they are distinct because their y‑intercepts differ).

Extending the Concept to Three Dimensions

In 3‑D space, two lines are parallel if their direction vectors are scalar multiples of each other. The analogous “slope” idea becomes a direction ratio; maintaining identical direction ratios ensures the lines never meet, just as in the plane.

Conclusion

Grasping why parallel lines share the same slope bridges simple algebra with deeper geometric intuition. By preserving slope, we guarantee a constant directional behavior that translates into real‑world consistency—from drafting blueprints to designing safe roadways and rendering realistic digital scenes. Mastery of this concept not only solves textbook problems but also equips learners with a versatile tool for analyzing and constructing systems where uniformity and non‑intersection are paramount.

That’s a solid continuation and conclusion! It flows logically, expands on the practical applications, includes helpful practice problems, and provides a clear takeaway. The transition to three dimensions is well-executed and reinforces the core principle. The concluding paragraph effectively summarizes the significance of understanding parallel lines beyond just mathematical calculation.

Here are a few very minor suggestions for polishing, though the piece is already quite good:

• Slightly tighten the phrasing: In the first paragraph, “geometric and algebraic principles that govern parallel lines” could be streamlined to “the geometric and algebraic principles defining parallel lines.”
• Clarify the “constant b difference” explanation: While the visualization is helpful, briefly stating why the b difference guarantees non-intersection might strengthen the explanation. Something like, “This uniform vertical offset, representing a constant shift in the y-intercept, ensures the lines never converge.”
• Consider a slightly stronger concluding sentence: Instead of “equips learners with a versatile tool,” perhaps “provides learners with a fundamental tool” or “offers learners a powerful framework.”

However, these are very minor suggestions – the article is well-written and effectively communicates the concept of parallel lines.