Through 4 2 Parallel To Y 3 4x 5

Introduction

The task is to find the equation of a line through 4 2 parallel to y 3 4x 5. In plain mathematics this means we need the equation of a line that passes through the point ((4,,2)) and has the same slope as the line given by (y = \frac{3}{4}x + 5). This article explains the concept of parallel lines, walks through each step required to derive the desired equation, highlights common pitfalls, and answers frequently asked questions. By the end, readers will be able to solve similar problems confidently and understand why the method works.

Understanding Parallel Lines

The concept of slope

In coordinate geometry, the slope of a line determines its steepness and direction. The slope (m) is defined as the change in (y) divided by the change in (x) ((\frac{\Delta y}{\Delta x})). Two non‑vertical lines are parallel if and only if they share the same slope That's the part that actually makes a difference. And it works..

Visual intuition

Imagine two roads that never intersect no matter how far they are extended. Their inclines are identical, which is why their slopes are equal. This visual cue helps students grasp why parallelism depends solely on slope, not on the intercept or the specific points the lines contain Turns out it matters..

Identifying the Given Information

The point ((4,,2))

The coordinates ((4,,2)) tell us the exact location through which the unknown line must pass. In the point‑slope formula, this point provides both the (x)‑ and (y)-values needed to anchor the line Turns out it matters..

The reference line (y = \frac{3}{4}x + 5)

The equation is already in slope‑intercept form (y = mx + b), where (m) is the slope and (b) is the (y)-intercept. But here, the slope (m = \frac{3}{4}). Because parallel lines share the same slope, the line we are looking for must also have a slope of (\frac{3}{4}) Simple, but easy to overlook..

Steps to Find the Equation

1. Determine the slope
   Extract the slope from the given line.
   Slope (m = \frac{3}{4}).

2. Use the point‑slope form
   The point‑slope formula is:
   [ y - y_1 = m(x - x_1) ]
   Substitute (m = \frac{3}{4}), (x_1 = 4), and (y_1 = 2) The details matter here..

3. Simplify to slope‑intercept form
   Expand and rearrange the equation to obtain (y = mx + b).

4. Verify the result
   Plug the point ((4,,2)) back into the derived equation to ensure it satisfies the line.

Each step is explained in detail below.

Detailed Walkthrough

Step 1 – Extract the slope

From (y = \frac{3}{4}x + 5) we see that the coefficient of (x) is the slope.
[ m = \frac{3}{4} ]
Italic note: The fraction (\frac{3}{4}) indicates that for every 4 units the line moves horizontally, it rises 3 units vertically.

Step 2 – Apply the point‑slope formula

Insert the known values:

[ y - 2 = \frac{3}{4},(x - 4) ]

This equation tells us that the line we want passes through ((4,2)) and has the same incline as the reference line.

Step 3 – Simplify to slope‑intercept form

1. Distribute the slope on the right‑hand side:

   [ y - 2 = \frac{3}{4}x - \frac{3}{4}\times4 ]

2. Calculate the product:

   [ \frac{3}{4}\times4 = 3 ]

   So the equation becomes

   [ y - 2 = \frac{3}{4}x - 3 ]

3. Add 2 to both sides to isolate (y):

   [ y = \frac{3}{4}x - 3 + 2 ]

4. Combine constants:

   [ y = \frac{3}{4}x - 1 ]

Thus the final equation of the line through 4 2 parallel to y 3 4x 5 is

[ \boxed{y = \frac{3}{4}x - 1} ]

Step 4 – Verification

Plug (x = 4) into the derived equation:

[ y = \frac{3}{4}\times4 - 1 = 3 - 1 = 2 ]

The result matches the given (y)-coordinate (2), confirming that the point lies on the line.

Common Mistakes and How to Avoid Them

• Misreading the point – Ensure the coordinates are correctly identified as ((4,2)) and not swapped (e.g., ((2,4))). A swapped point leads to an incorrect line.
• Forgetting the sign of the slope – The slope (\frac{3}{4}) is positive; a negative sign would produce a line that rises in the opposite direction.
• Algebraic errors during distribution – Carefully multiply (\frac{3}{4}) by each term inside the parentheses.
• Skipping verification – Always substitute the original point back into the final equation; this quick check catches most mistakes.

FAQ

What does “parallel” really mean?

Parallel lines are lines in the same plane that never intersect. Mathematically, this occurs when their slopes are equal while their (y)-intercepts differ (or, in the special case of vertical lines, both are vertical) Simple as that..

Can a vertical line be parallel to a non‑vertical line?

No. Also, a vertical line has an undefined slope, while any non‑vertical line has a defined finite slope. Since the slopes cannot be equal, they cannot be parallel Simple, but easy to overlook. Less friction, more output..

Why do we use the point‑slope form instead of the slope‑intercept form directly?

The point‑slope form (y - y_1 = m(x - x_1)) directly incorporates a known point and the slope, making it the most efficient way to construct the equation when a specific point is given The details matter here. Worth knowing..

What if the given line is already in standard form (Ax + By = C)?

First, rearrange it to slope‑intercept form to identify the slope (m). Then follow the same steps: extract (m), apply point‑slope, and simplify.

Is the y‑intercept of the new line always different from the original?

Not necessarily. Still, if the original line already passes through the given point, the intercepts will be identical. Otherwise, the intercept will differ, as seen in our example where the original intercept is 5 and the new intercept is (-1).

Conclusion

Finding the equation of a line through 4 2 parallel to y 3 4x 5 is a straightforward application of the principle that parallel lines share the same slope. By extracting the slope from the reference line, using the point‑slope formula with the given coordinates, and simplifying to slope‑intercept form, we obtain the precise equation (y = \frac{3}{4}x - 1). Understanding each step, watching out for common errors, and verifying the result ensure confidence in solving similar problems. Mastery of this technique builds a solid foundation for more advanced topics in coordinate geometry and algebra.

To solidify the procedure, try a different set of numbers. Suppose you need the equation of the line that passes through ((-1,,5)) and is parallel to the line (y = -\frac{2}{3}x + 7) It's one of those things that adds up..

1. Extract the slope.
   The reference line is already in slope‑intercept form, so the slope is (-\frac{2}{3}). Because parallel lines share the same slope, the new line must also have (-\frac{2}{3}) Worth knowing..

2. Apply the point‑slope formula.
   [ y - 5 = -\frac{2}{3}\bigl(x - (-1)\bigr) ;=; -\frac{2}{3}(x + 1) ]

3. Simplify to slope‑intercept form.
   Distribute the slope carefully: [ y - 5 = -\frac{2}{3}x - \frac{2}{3} ] Add 5 (or (\frac{15}{3})) to both sides: [ y = -\frac{2}{3}x - \frac{2}{3} + \frac{15}{3} ] [ y = -\frac{2}{3}x + \frac{13}{3} ]

4. Verification step.
   Substitute the original point ((-1,5)): [ 5 ; \stackrel{?}{=} ; -\frac{2}{3}(-1) + \frac{13}{3} ] The right‑hand side equals (\frac{2}{3} + \frac{13}{3} = \frac{15}{3} = 5), confirming the equation is correct.

Additional tips

• When the given line is in standard form (Ax + By = C), first rewrite it as (y = mx + b) to read the slope directly.
• If the point’s coordinates are fractions, clear the denominators early to avoid cumbersome arithmetic.
• Always double‑check the sign of the slope; a missed negative will send the line in the opposite direction.
• After simplifying, a quick substitution of the original point serves as a reliable sanity check.

By consistently extracting the correct slope, using the point‑slope template, and confirming the result, you can handle any parallel‑line problem with confidence. Regular practice of these steps turns a potentially error‑prone task into a routine calculation, paving the way for more advanced work in coordinate geometry and algebraic modeling That's the part that actually makes a difference. Which is the point..