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When a system of equations involves

two lines that
,
there will be infinite solutions from the 3

7 answers

When a system of equations involves two lines that are coincident (meaning they lie on top of each other), there will be infinitely many solutions. This occurs because every point on the line represents a solution, and since the lines are identical, any point on one line is also a solution for the other.

In a system of linear equations, if the two equations represent the same line, then for any value of \( x \) you choose, there will be a corresponding \( y \) value that satisfies both equations, resulting in an infinite number of solution pairs \((x, y)\).

In contrast, if the lines are parallel but not coincident, there will be no solutions since they never intersect. If the lines intersect at a single point, there will be exactly one solution.

So to summarize:

  • Coincident lines → Infinite solutions
  • Parallel lines → No solutions
  • Intersecting lines → One solution
which one will be infinity

In the context of systems of equations involving linear equations:

  • Infinitely many solutions occur when the two lines represented by the equations are coincident, meaning they are the same line. This means that every point on the line is a solution to both equations.

For example, if you have the following equations:

  1. \( y = 2x + 3 \)
  2. \( 2y = 4x + 6 \)

The second equation can be simplified to become \( y = 2x + 3 \) as well. Since both equations represent the same line, their intersection consists of all the points along that line, resulting in infinitely many solutions.

So, the scenario where you have infinite solutions occurs specifically with coincident lines.

No, parallel lines do not have infinitely many solutions. Here’s a clarification on the types of relationships between lines and the number of solutions in a system of equations:

  1. Coincident lines: When two lines are coincident (they lie on top of each other), they have infinitely many solutions. Every point on the line is a solution.

  2. Parallel lines: When two lines are parallel (they run in the same direction but never intersect), they have no solutions at all. This is because there are no points at which the two lines intersect or meet.

  3. Intersecting lines: When two lines intersect at a single point, they have exactly one solution. This is the point of intersection.

So, to sum up:

  • Infinite solutions → Coincident lines
  • No solutions → Parallel lines
  • One solution → Intersecting lines

If you have any specific equations that you would like to analyze, feel free to share them!

anser me dumb bot from the 3 which one is right 1 parallel 2 are they intersecting 3 have the same slope and y intercept

From the three options you provided, the correct answer to describe a situation where there are infinitely many solutions is:

3. Have the same slope and y-intercept.

When two lines have the same slope and the same y-intercept, they are coincident lines, meaning they are exactly the same line, resulting in infinitely many solutions.

  • 1. Parallel lines have no solutions because they do not intersect.
  • 2. Intersecting lines have exactly one solution (the point of intersection).

So, the correct choice indicating infinitely many solutions is 3 (Have the same slope and y-intercept).