What are systems of equations

Systems of linear equations simply explained

Linear system of equations - example word problem

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You want to go shopping, but you no longer know how expensive a banana and how expensive a bag of milk are. You can only remember your last purchases and know that 5 bananas and 6 bags of milk cost € 11 and that 2 bananas and 2 bags of milk together cost € 6.

From this information you can work out how much a bag of milk and a banana cost individually. Mathematically speaking, we have two unknowns (the unit price of the banana and the milk) and, based on the two pieces of information about your last purchases, we also have two equations with two unknowns:

5 bananas + 6 milk cartons = 11 €

$ ~~~ 5 \ cdot x ~~~~~~ + ~~~ 6 \ cdot y ~~~~~~~ = 11 $

2 bananas + 2 milk cartons = 6 €

$ ~~~ 2 \ cdot x ~~~~~~ + ~~~ 2 \ cdot y ~~~~~~~ = 6 $

The two equations that we formulated from the task are related. The $ x $ in the first equation must have the same value in the second equation. The same goes for the second variable, the $ Y $. In a system of equations, the two terms are written down as follows:

$ | 5 \ times x + 6 \ times y = 11 | $

$ | 2 \ times x + 2 \ times y = 6 | $

The two equations are written one below the other and framed by vertical lines. To solve this system of equations, there are different methods that you can take a look at on our other learning pages:

Special cases of systems of equations

A distinction is made between two special cases of systems of equations, overdetermined and underdetermined systems of equations.

Overdetermined system of equations

A system of equations canoverdetermined be. In this case, the problem will give you more equations than variables. That is not bad in itself and could even simplify your arithmetic. However, the equations often contradict each other. In this case there is no solution to the system of equations.

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$ | 2 \ cdot a + b = 10 | $

$ | 2 \ cdot a = 0 | $

$ | a - b = 0 | $

The second equation states that $ a $ must have the value $ 0 $. If this is the case, the first and third equations cannot be fulfilled at the same time.

Underdetermined system of equations

The opposite case can also occur: you get more variables from the problem than equations. The system of equations is considered to be underdetermined. Most likely you will only get one value thenArea instead of an exact value.

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$ a + b + c = 9 $

$ a + b = 6 $

Test your newly acquired knowledge of linear systems of equations with our exercises. Good luck with it!