Enter a line equation (in slope-intercept or standard form) and a point (x₀, y₀) to find the equation of a parallel line passing through that point. You'll get the parallel line equation in slope-intercept form, the shared slope, the y-intercept, and the distance between the two lines.
Parallel Line Equation
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Shared Slope (a)
New Y-Intercept (b₂)
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Distance Between Lines
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Parallel lines always have the same slope. If the original line has equation y = ax + b, the parallel line also has slope a. You only need to find the new y-intercept so the parallel line passes through your desired point.
First, identify the slope (a) of the original line. Since the parallel line has the same slope, write y = ax + b₂. Then substitute the coordinates of your point (x₀, y₀) into this equation and solve for b₂: b₂ = y₀ − a·x₀. The result is the full equation of the parallel line.
For two parallel lines y = ax + b₁ and y = ax + b₂, the distance is |b₂ − b₁| / √(a² + 1). This formula accounts for the slope of the lines — horizontal lines (slope = 0) simplify the formula to just |b₂ − b₁|.
Two lines are parallel if and only if they have the same slope (coefficient of x) but different y-intercepts. If the y-intercepts are also equal, the lines are identical, not parallel.
The y-intercept is the point where a line crosses the y-axis, i.e. where x = 0. In the slope-intercept form y = ax + b, the constant b is the y-intercept value.
Yes. Vertical lines have undefined slope and are written as x = c. Any other vertical line x = d (where d ≠ c) is parallel to it. This calculator handles non-vertical lines in slope-intercept form y = ax + b.
Parallel lines appear everywhere: railway tracks, the edges of a ruler, lined notebook paper, lanes on a road, and the opposite sides of a rectangle are all examples of parallel lines that never intersect and maintain a constant distance apart.
A parallel line has the same slope as the original line, while a perpendicular line has a slope that is the negative reciprocal. For a line with slope a, a parallel line has slope a and a perpendicular line has slope −1/a.