## parallel lines

### Parallel lines are a pair of lines that never intersect or cross each other, no matter how far they are extended

Parallel lines are a pair of lines that never intersect or cross each other, no matter how far they are extended. They have the same slope but different y-intercepts. In other words, their gradients (or slopes) are equal, while their y-intercepts are different.

To determine if two lines are parallel, we need to compare their slopes. If the slopes of the two lines are equal, then the lines are parallel. Alternatively, if the lines have different slopes, they are not parallel.

For example, let’s say we have two lines, line A with equation y = 2x + 3 and line B with equation y = 2x – 1. We can compare their slopes:

– The slope of line A is 2.

– The slope of line B is also 2.

Since the slopes of both lines are equal (2), we can conclude that line A and line B are parallel.

Another way to determine if two lines are parallel is by comparing their equations. If the equations of two lines have the same form (such as y = mx + b), then the lines are parallel. However, it’s important to note that having the same form does not guarantee parallelism. The slopes must still be equal.

For instance, consider line C with equation y + 3x = 7 and line D with equation y + 3x = 10. Although they have the same form, if we rearrange the equations, we can find the slopes:

– Line C: y + 3x = 7 can be rearranged as y = -3x + 7 with a slope of -3.

– Line D: y + 3x = 10 can be rearranged as y = -3x + 10 with a slope of -3.

Since the slopes of both lines are equal (-3), we can conclude that line C and line D are parallel as well.

In summary, parallel lines are never crossing lines. They have the same slope but different y-intercepts, or they have equations in the same form with equal slopes. Remember, comparing slopes is the most reliable method to determine whether lines are parallel.

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