Yes, a system of linear equations can have one unique solution.
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Yes, a system of linear equations can have one unique solution. This occurs when the equations are independent, meaning that none of them can be derived from any combination of the others.
A famous quote related to linear equations is from mathematician John von Neumann, who said: “In mathematics you don’t understand things. You just get used to them.” This rings true for many students learning about linear equations for the first time, as they can seem quite abstract.
Here are some interesting facts related to linear equations:
- Linear equations have been studied for thousands of years, dating back to ancient Greece and China.
- The ancient Babylonians also had a method for solving linear equations, which involved breaking down the equation into simpler parts and solving each part separately.
- The concept of matrix algebra, which is used to solve systems of linear equations, was developed in the 1800s by mathematicians such as Augustin-Louis Cauchy and James Joseph Sylvester.
- Linear equations have practical applications in fields such as engineering, economics, and physics. For example, they can be used to model how a system will behave over time.
- One of the most famous applications of linear equations is in linear regression, which is used to find a line of best fit for data points. This is commonly used in fields such as statistics and machine learning.
To better understand how a system of linear equations can have one solution, let’s look at an example. Consider the following system of equations:
2x + 3y = 5
4x – 2y = 6
To solve this system, we can use the method of elimination. Multiplying the first equation by 2 and subtracting it from the second equation gives:
-8y = -4
y = 1/2
Substituting this value into the first equation gives:
2x + 3(1/2) = 5
2x = 4
x = 2
So the unique solution to the system is x = 2 and y = 1/2. This occurs because the equations are independent; one cannot be derived from the other.
Here is a table summarizing the possible outcomes for a system of linear equations:
|Number of Solutions||Type of System|
Overall, while linear equations can seem daunting at first, they are a powerful tool with many applications in the real world.
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.
Many additional responses to your query
A system of linear equations usually has a single solution, but sometimes it can have no solution (parallel lines) or infinite solutions (same line).
A system of linear equations can have one solution if there is a single point that makes every equation in the system true. This means that there is one point that can satisfy all of the equations at the same time.
A system of linear equations has one solution when the graphs intersect at a point.
Every linear system of equations has exactly one solution, infinite solutions, or no solution. This leads us to a definition. Here we don’t differentiate between having one solution and infinite solutions, but rather just whether or not a solution exists.
Solutions to a system of linear equations.A system of linear equations can haveno solutions, exactly one solution, or innitely many solutions. If the system has twoor more distinct solutions, it must have innitely many solutions.
A solution of a linear system is an assignment of values to the variables x1, x2,…, xn such that each of the equations is satisfied. The set of all possible solutions is called the solution set. A linear system may behave in any one of three possible ways: The system has infinitely many solutions. The system has a single unique solution.
A consistent system of equations has at least one solution. A consistent system is considered to be an independent system if it has a single solution, such as the example we just explored.
However, systems of two linear equations with two variables can have a single solution that satisfies both solutions.
With linear systems of equations, there are three possible outcomes in terms of number of solutions:
• One solution.
• Infinitely many solutions.
• No Solutions at all.
If there is one solution, it means that there is a single intersection between the two lines that your equations give. For example, considering the following equations:
If we graphed them both, this is what we’d end up with:
made on desmos online graphing calculator
Blue line: y=−x+9
Red line: y=2x
As you can see, both lines intersect at the coordinate (3,6). This means that (3,6) is a solution to both equations, or in simpler terms, if you plugged in 3 for x in either equation, you would get 6 for y.
If there are infinitely many solutions, it means that both your equations are referring to the same line . For example, if we took our graph from last time but made both lines y=2x:
made on desmos online graphing calculator
As you see, the two lines are indiscernible from one another. Hence we can sa…
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