The graph of two lines intersecting at one point shows a system of equations with exactly one solution.

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The graph of two lines intersecting at one point shows a system of equations with exactly one solution. As mathematician John von Neumann said, “In mathematics you don’t understand things. You just get used to them.” However, there are some interesting facts to know about systems of equations and their graphs:

- A system of equations is a set of two or more equations with the same variables. The solution to the system is the set of values that makes all equations true.
- There can be three types of solutions to a system of equations: no solution, infinite solutions, and exactly one solution.
- When graphing two linear equations, there are three possible scenarios: the lines intersect at one point (one solution), the lines are parallel (no solution), or the lines coincide (infinite solutions).
- The graph of a system of linear equations with one solution is called consistent and independent. This means that the equations are not multiples of each other and do not reduce to a single equation.
- In terms of matrices, a system of linear equations with one solution has a coefficient matrix that is invertible.
- Other systems of equations, such as quadratic or nonlinear equations, can also have exactly one solution, but their graphs may not be straight lines.

Here is a table summarizing the possible solutions of a system of two linear equations:

Type of solution | Description | Graph |
---|---|---|

No solution | The lines are parallel and do not intersect. | |

Infinite solutions | The lines coincide and are the same line. | |

Exactly one solution | The lines intersect at one point. |

In conclusion, the graph of two lines intersecting at one point shows a system of equations with exactly one solution. While solving systems of equations can be challenging, graphing them can provide a visual representation that helps understand the solutions. As mathematician Jacob Bronowski said, “The principle of science, the definition, almost, is the following: The test of all knowledge is experiment. Experiment is the sole judge of scientific ‘truth.'”

## Response via video

The YouTube video “One Solution, No Solution, or Infinitely Many Solutions – Consistent & Inconsistent Systems” explains how to determine if a system of equations is consistent or inconsistent, dependent or independent, and contains one solution, no solution, or many solutions. By solving the system of equations, a single value for x and y indicates one solution, a contradiction shows no solution, and a statement like 0 = 0 or x = x means many solutions. The video also shows examples and uses the elimination method to obtain equations that indicate whether the system is consistent, dependent, or independent.

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If the graphs of the equations intersect, then there is one solution that is true for both equations. If the graphs of the equations do not intersect (for example, if they are parallel), then there are no solutions that are true for both equations.

Answer:The last graph, where the functions intersect at only one point.Step-by-step explanation:A system of equations with only one solution is the one that in some point the solutions of each ecuation is exactly the same. In the graphs 1 and 3, where the equations does not intersect each other, the system of equations have no solution. In the second graph, the equations intersect 2 times, so the system of equations have 2 solutions.

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Also Know, **Which system of equations has exactly one solution?** Response: independent system

An independent system has exactly one solution pair (x,y) . The point where the two lines intersect is the only solution.

**Which graph most likely shows a system of equations with one?** The answer is: The graph of a system with one solution is **two intersecting lines**.

In respect to this, **What is a graph that shows no solution?**

The response is: A graph with no solution will have **functions that do not all intersect at any point**. If the system consists of two functions, then there will be no points of intersection. A system of two linear equations has no solution if the lines are parallel.

Subsequently, **How to tell if a system of equations has one solution without graphing?**

1 Answer. A system of N linear equations with N unknown variables that contains no linear dependency between equations (in other words, its determinant is non-zero) will have one and only one solution.