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.

## 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 $\pounds 100.$ 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 $\pounds 120$ 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 $\pounds 100$ total, therefore...

$$$3x+y=100$

Similarly, we are told that two child's tickets and 2 adult's tickets cost $\pounds \mathrm{120\; in}$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.

$2x+2y=120$

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

$x=Child\text{'}sticket=\pounds 20$

$y=Adult\text{'}sticket=\pounds 40$

## 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.

$\left\{\begin{array}{l}y=4x-1\\ y=x+2\\ y=2x+1\end{array}\right.$

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.

$\left\{\begin{array}{l}y=x+1\\ y=x+1\end{array}\right.$

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 of$x$and $y$for which all three equations hold true.

$\left\{\begin{array}{l}y=4x-3\\ y=x+2\\ y=2x+1\end{array}\right.$

## Linear Systems Examples

Which of the following are linear systems?

**(a)**

$\left\{\begin{array}{l}y=2x+4\\ y=3{x}^{2}-5\\ y=4x+3\end{array}\right.$

**(b)**

**$\left\{\begin{array}{l}y=3x+1\\ y=5x-2\\ y=x+3\end{array}\right.$**

**(c)**

**$\left\{\begin{array}{l}y=3\\ y=4x+2\\ y=x-5\end{array}\right.$**

**(d)**

**$\left\{\begin{array}{l}y=4x\\ y=x-10\\ y=6x+3\\ y=2x-12\\ y=4x+12\\ y=3x-2\end{array}\right.$**

**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)**

**(b)**

**(c)**

**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.

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##### Frequently Asked Questions about Linear Systems

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

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.

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