How Many Solutions Exist When Two Lines Share The Same Slope And Y-Intercept?

What happens when two lines share both the same slope and the same y-intercept? When two lines have identical slopes and y-intercepts (for example, y = (5/4)x + 1 and y = (5/4)x + 1), it results in an infinite number of solutions. In such cases, these lines overlap perfectly along their entire length, essentially representing the same line in the coordinate plane.

On the other hand, what occurs when two lines share the same slope but different y-intercepts? When two lines have matching slopes but distinct y-intercepts (for instance, y = 2x + 1 and y = 2x – 3), there are no common solutions. This means that these lines are parallel to each other and will never intersect, indicating that there are no points where they cross paths.

Please note that the date “12th July 2018” at the end of the original passage appears to be unrelated to the topic and can be omitted unless it has specific relevance to the context.

How many solutions are there when the slopes of two linear equations are the same? The number of solutions depends on whether the equations represent the same line or two different lines with the same slope.

1. Coincident Lines (Infinite Solutions): When the equations represent the same line, they are coincident. In this case, there are an infinite number of solutions because all points on the line satisfy both equations simultaneously.

2. Parallel Lines (No Solution): If the equations have the same slope but do not represent the same line, they are parallel. Parallel lines will never intersect, leading to the conclusion that there are no solutions to the system of equations in this scenario.

Solution for Lines with the Same Slope:

When two lines share the same slope, there are a few possible scenarios to consider. Firstly, these lines could be coincident, which means they overlap entirely and have all their points in common. Alternatively, they might be parallel, in which case they will not share any common points. To have precisely one point of intersection, Line 2 must possess a slope distinct from that of Line 1, which is crucial because identical slopes would lead to either coincident or parallel lines, as described above. This distinction in slopes allows the two lines to intersect at a single point, creating a unique solution.