A system of linear equations with exactly one solution is called a consistent and independent system.
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A system of linear equations with exactly one solution is called a consistent and independent system. This means that the system has a unique solution and the equations are not dependent on each other. According to famous mathematician, Euclid, “The laws of nature are but the mathematical thoughts of God.” Linear equations are essential in various fields such as engineering, physics, economics, and finance. Here are some interesting facts related to linear equations:
- Linear equations were first introduced by Islamic mathematicians in the 9th century.
- The number of equations should be equal to or greater than the number of variables in a linear system.
- In a linear system, the solution set can be represented as a subset of Euclidean space.
- Cramer’s rule states that a linear system with n equations and n variables has a unique solution if its determinant is nonzero.
- A linear system can be solved manually using elimination or substitution methods, or through matrix operations.
- The concept of linear independence is crucial in solving systems of linear equations.
- A system of linear equations with no solutions is called an inconsistent system.
Below is a table showing some examples of consistent and independent systems of linear equations:
| System of Linear Equations | x-value | y-value |
|---|---|---|
| 2x + y = 5 3x – y = 7 |
2 | 1 |
| x + 2y = 9 4x – y = 10 |
2 | 3 |
| 2x + y – z = 6 4x – y + z = 2 6x + 2y – 3z = 15 |
1 | 2 |
| 3x – y = 5 6x – 2y = 10 9x – 3y = 15 |
5 | 15/2 |
In conclusion, a consistent and independent system of linear equations is characterized by a unique solution. Understanding linear equations is crucial in solving problems that involve numerical relationships in various fields.
Response to your question in video format
This YouTube video explains the three possible solutions for a linear system: a unique solution, no solution, or infinitely many solutions. The speaker uses the matrix Ax=b to demonstrate how to solve a system of equations and explains the importance of understanding the number of rows and columns in the matrix. They provide examples of each solution scenario and highlight situations where there is no solution or infinitely many solutions. The video also notes that understanding the variables X, K, and B is crucial in solving linear systems.
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A linear system that has exactly one solution is called a consistent independent system. Consistent means that the lines intersect and independent means that the lines are distinct.
You will most likely be intrigued
Similarly one may ask, What is system of two linear equations with one solution?
As a response to this: If the two lines have two different slopes, then they will intersect once. Therefore, the system of equations has exactly one solution. If the two lines have the same slope but different y-intercepts, then they are parallel lines, and they will never intersect.
What does it mean when a system of equations only has one solution?
Equations with one solution
Some equations have exactly one solution. In these equations, there is only one value for the variable that makes the equation true. You can tell that an equation has one solution if you solve the equation and get a variable equal to a number.
Herein, Which system of linear equations has a solution of one 1? As an answer to this: A consistent linear system of equations will have exactly one solution if and only if there is a leading 1 for each variable in the system. If a consistent linear system of equations has a free variable, it has infinite solutions.
Thereof, Can a system of two linear inequalities have exactly one solution? Answer will be: System of inequalities may have zero, one or many solutions. Zero solution means there is not a single ordered pair which satisfies both the inequalities. One solution means only one ordered pair (x,y) falls in the solution area of the graph.
Similarly one may ask, What does a system of equations mean?
Answer to this: A system of equation just means ‘mmore than 1 equation.’. A system of linear equations is just more than 1 line, see the picture: Ok, so what is the solution of a system of equations? The solution is where the equations ‘meet’ or intersect. The red point is the solution of the system. How many solutions can systems of linear equations have?
Secondly, Is there a solution to a linear equation?
Answer to this: There are no points common to both lines; hence, there is no solution to the system. There are three types of systems of linear equations in two variables, and three types of solutions. An independent system has exactly one solution pair (x,y) ( x, y). The point where the two lines intersect is the only solution.
What is a system of two linear equations in two variables?
The reply will be: A system of two linear equations in two variables has one solution when the two lines have different slopes. From an algebra standpoint, this means that we get a single value when solving the system. Visually, the lines intersect exactly once on a graph, since they have different slopes.
In this way, Does a linear system have a unique solution? The answer is: Some linear systems may not have a solution and others may have an infinite number of solutions. In order for a linear system to have a unique solution, there must be at least as many equations as there are variables. Even so, this does not guarantee a unique solution.