Parallel Line Calculator

Find the equation of a line parallel to a given line through a specific point

Original Line

Line Equation (y = mx + b format)

Point for Parallel Line

X-coordinate

Y-coordinate

Results

Original Line Equation:

y = 2x + 3

Parallel Line Equation:

y = 2x + 3

Slope (m):

2

Y-intercept (b):

3

How It Works

Parallel lines have identical slopes. The calculator finds the slope of your original line, then calculates the new y-intercept (b) using the formula: b = y - mx, where (x,y) is your given point.

A parallel line calculator helps determine the equation of a line that runs parallel to a given line while passing through a specified point. This is useful in geometry, engineering, and computer graphics.

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🔹 How to Find a Parallel Line

Two lines are parallel if they have the same slope (m) but different y-intercepts (b).

Steps to Calculate a Parallel Line

1. Identify the slope (m) of the original line.

2. Use the same slope (m) for the parallel line.

3. Plug in the new point (x₁, y₁) into the point-slope form:

   $$y-y_1=m(x-x_1)$$

4. Convert to slope-intercept form (y = mx + b) if needed.

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🔹 Example Calculation

Given Line:  y=2x+3
Point for New Line: (4, 5)

Step 1: Original slope $m=2$
Step 2: Parallel slope remains $m=2$
Step 3: Plug into point-slope form:

$$y-5=2(x-4)$$

Step 4: Simplify to slope-intercept form:

$$y=2x-8+5$$$$y=2x-3$$

Result: The parallel line equation is $y=2x-3$.

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🔹 Special Cases

• Vertical Lines (x = a) → Parallel line is  $x=\text{new x-intercept}$.
• Horizontal Lines (y = b) → A Parallel line is  $y=\text{new y-intercept}$.

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🔹 Manual Calculation (Step-by-Step)

1. Given Line: $3x+4y=8$

2. Find Slope (m):

   $$4y=-3x+8\to y=-\frac{3}{4}x+2\to m=-\frac{3}{4}$$

3. New Point: (1, -2)

4. Point-Slope Form:

   $$y-(-2)=-\frac{3}{4}(x-1)$$

5. Simplify:

   $$y+2=-\frac{3}{4}x+\frac{3}{4}$$$$y=-\frac{3}{4}x-\frac{5}{4}$$

Final Equation: $y=-\frac{3}{4}x-\frac{5}{4}$

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🔹 Applications

✔ Engineering (Designing parallel structures)
✔ Architecture (Aligning walls/edges)
✔ Game Development (Pathfinding algorithms)

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🚀 Try It Yourself!

Given Line: $y=-5x+1$
New Point: (2, 4)
Can you find the parallel line equation?

Answer:

$$y=-5x+14$$