Linear Systems

Dinner is booked for half an hour after the movie. When then, is dinner booked for? It's impossible to know, you don't have enough information! What if, however, you are told that the movie ends at six o'clock? Well, then you probably know intuitively that dinner must be at half-past six. You have just instinctively solved a linear system, and probably didn't even realise!

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    A linear system is a group of linear equations involving the same variables. There is no limit to the number of equations a linear system can have.

    These linear systems are an important aspect of mathematics, that can be used to describe real-world scenarios as well as more abstract problems such as in linear algebra. In this article, you'll learn how to use linear equations to build linear systems, and how to solve such systems.

    Let's consider the equations y=x+2 and y=2x+1 ,together they form a linear system. By plotting each equation on a graph, we can get a visual overview of the linear system as a whole.

    Linear Systems, Linear systems example graph, StudySmarterExample of a linear system, John Hannah - StudySmarter Originals

    How to Build a Linear System

    Often, we might be presented with a problem or real-world scenario that is actually a linear system. If we can recognize that a linear system is being described, and have the correct information at our disposal, then we can build it by expressing it algebraically. Let's take a look at an example to see how this is done.

    A woman buys concert tickets for her three children, as well as an adult ticket for herself. The total cost of her tickets is £100. Her friend buys a ticket each for himself and his spouse to the same concert, as well as two tickets for their children. Her friend paid £120 in total for his tickets.

    Solution:

    Firstly, how can we recognize that this is a linear system? Well, the observant may notice that there are two variables in the scenario, common to both purchasers: the cost of a child's ticket and the cost of an adult's ticket. After all, a linear system is just a group of linear equations involving the same variables.

    Now, how do we actually build our linear system from this information? We start by labelling each variable we have discerned. Let's say that x is the cost of a child's ticket, and y is the cost of an adult's ticket. From here we simply construct two equations from the information above.

    The information we are given says three child's tickets and one adult's cost £ 100 total, therefore...

    3x+ y=100

    Similarly, we are told that two child's tickets and 2 adult's tickets cost £ 120 in total, therefore...

    2x+2y=120

    And with those equations, we have just built our first linear system!

    3x+ y=100

    2x+ 2y=120

    How to Solve Linear Systems

    Linear systems are useful because they can be solved. For instance, these solutions can be used to tell when one runner in a race might overtake the other, or how much was paid each for an apple and banana at the shop.

    A solution to a linear system is the assignment of values to each variable such that all equations in the system hold true. When visualized on a graph, this is the point where the lines of all of the equations cross.

    Let's consider the linear system we just built concerning the adults' and children's concert tickets.

    3x+y=100

    2x+ 2y=120

    The solution to this system, presented as an ordered pair, is (20, 40). This communicates that a child's ticket costs £20, and an adult's costs £40. Try plugging these values in for x and y to prove that the equations hold true.

    x = Child's ticket = £20

    y = Adult's ticket = £40

    Types of Linear Systems

    Any linear system can be categorized as one of two types depending on the number of solutions it possesses; it is said to be consistent or inconsistent.

    A linear system is said to be consistent if it has one or more solutions. Furthermore, a dependent linear system has infinite solutions, and an independent system has a unique solution.

    The following linear system is said to be consistent and independent as it has a unique solution, which is (1, 3). We can tell that this system has a solution as we can clearly see that there is one point where all three lines intersect.

    y = 4x - 1y = x + 2y = 2x + 1

    Linear Systems, linear systems graph of consistent independent linear system, StudySmarterGraph of a consistent, independent linear system, John Hannah - StudySmarter Originals

    The following linear system is said to be consistent and dependent, as it has an infinite number of solutions. When graphed it appears as a single line, however, there are two equations in this system and they are equal at all points.

    y = x + 1y = x + 1

    Linear Systems, linear systems Graph of a consistent, dependent linear system, StudySmarterGraph of a consistent, dependent linear system, John Hannah - StudySmarter

    A linear system is said to be inconsistent if it has no solutions.

    The following linear system is said to be inconsistent as it has no solution. We can tell that this system has no solution as we can clearly see that there isn't a point where all three lines intersect. This is because there are no values ofxand yfor which all three equations hold true.

    y = 4x -3y = x + 2y = 2x + 1

    Linear Systems, linear systems Graph of an inconsistent linear system, StudySmarterGraph of an inconsistent linear system, John Hannah - StudySmarter Originals

    Linear Systems Examples

    Which of the following are linear systems?

    (a)

    y = 2x + 4y = 3x2 -5y = 4x +3

    (b)

    y = 3x +1y = 5x -2y = x + 3

    (c)

    y = 3y = 4x + 2y = x-5

    (d)

    y = 4xy = x -10y = 6x + 3y = 2x-12y = 4x +12y = 3x - 2

    Solutions:

    (a) Not linear system (b) Linear system (c) Linear system (d) Linear system

    Classify the following linear systems as dependent consistent, independent consistent, or inconsistent.

    (a)

    Linear Systems, linear systems example inconsistent, StudySmarter Types of linear systems example, John Hannah - StudySmarter Originals

    (b)

    Linear Systems, linear systems example consistent, StudySmarter Types of linear systems example, John Hannah - StudySmarter Originals

    (c)

    Linear Systems, linear systems example consistent, StudySmarter Types of linear systems example, John Hannah - StudySmarter Originals

    Solutions:

    (a) Inconsistent (b) Consistent, Independent (c) Consistent, dependent

    Linear Systems - Key takeaways

    • Linear systems are collections of linear equations that share the same variables.
    • There is no limit to the number of equations or variables these linear systems can contain.
    • Given enough information, it is possible to build a linear system from a real-world scenario.
    • A solution to a linear system is the assignment of values to each variable such that all equations in the system hold true.
    • A linear system has can one solution, an infinite number of solutions, or no solution.
    Frequently Asked Questions about Linear Systems
    What are linear systems?

    Linear systems are collections of linear equations with the same variables.

    What is an example of a system of linear equations?

    An example of a linear system is:

                         y = 2x + 1

                         y = 3x + 2

    How do you solve linear systems?

    Linear systems can be solved using simultaneous equations, matrices, linear combinations, or graphs.

    What is the solution to a linear system?

    A solution to a linear system is the assignment of values to each variable such that all equations in the system hold true.

    What are types of linear systems?

    Linear systems can be categorised as consistent or inconsistent. A linear system is consistent if it has at least one solution and inconsistent if it has no solutions.

    Test your knowledge with multiple choice flashcards

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