Linear Systems

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.

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.

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 $\pounds 20,$ 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.

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.

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

Graph 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.

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

Graph of a consistent, dependent linear system, John Hannah - StudySmarter

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.

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

Graph of an inconsistent linear system, John Hannah - StudySmarter Originals

Linear Systems Examples