Yes, two different linear functions can have the same y-intercept. The y-intercept is the point where a line crosses the y-axis, and multiple lines can intersect the y-axis at the same point if they have different slopes. For example, the functions (y = 2x + 3) and (y = -1x + 3) both have a y-intercept of 3 but different slopes, making them distinct linear functions.
yes
No because only co-linear lines lie on the same plane
When two linear functions share the same rate of change, their graphs will be parallel lines because they have the same slope. However, their equations will differ in the y-intercept, which means they will cross the y-axis at different points. Consequently, their tables of values will show consistent differences in their outputs for the same inputs. Despite having the same slope, these differences lead to distinct linear functions.
-- An infinite number of different planes can intersect the same line. -- The same line can lie in an infinite number of different planes. -- An infinite number of different lines can intersect the same plane.
They are the same.
yes
No. A linear equation represents a straight line and the solution to a set of linear equations is where the lines intersect; two straight lines can only intersect at most at a single point - two straight lines may be parallel in which case they will not intersect and there will be no solution. With more than two linear equations, it may be that they do not all intersect at the same point, in which case there is no solution that satisfies all the equations together, but different solutions may exist for different subsets of the lines.
No because only co-linear lines lie on the same plane
Two lines intersect at one point. If in two dimensions, and they do not intersect they are parallel. The other option in two dimensions is they are the co-linear, that is they are the same line, in which case they intersect at all points.
Two linear equations (or lines) with the same y-intercept and different slopes are intersecting lines. They intersect at the y-intercept. If the slopes are negative reciprocals (ex: one slope is 3 and one slope it -1/3) then they are perpendicular lines.
-- An infinite number of different planes can intersect the same line. -- The same line can lie in an infinite number of different planes. -- An infinite number of different lines can intersect the same plane.
If the equations are linear, they may have no common solutions, one common solutions, or infinitely many solutions. Graphically, in the simplest case you have two straight lines; these can be parallel, intersect in a same point, or actually be the same line. If the equations are non-linear, they may have any amount of solutions. For example, two different intersecting ellipses may intersect in up to four points.
They are the same.
Lines in the same plane that do not intersect Lines in the same plane that do not intersect Lines in the same plane that do not intersect Lines in the same plane that do not intersect
Only if the two functions really represent the same function.
They are parallel lines which have the same slope but different y intercepts.
They are parallel lines which have the same slope but different y intercepts.