The type of container is not too important but the ones included in Bearing Cleaner Kits often stack bearings vertically and can be very useful for separating debris. Pour enough cleaner into your container so that your bearings will be completely submerged.

This is called an augmented matrix. The word “augmented” refers to the vertical line, which we draw to remind ourselves where the equals sign belongs; a matrix is a grid of numbers without the vertical line.

An augmented matrix corresponds to an inconsistent system of equations if and only if the last column (i.e., the augmented column) is a pivot column.

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Many bearings utilize rubber shields that pop right out but some have alloy shields that are locked into place with a C-clip. Use a safety pin or thumb tack to pry the shields (or C-clips) away from the bearing. Wipe each shield clear of debris and set them aside. Removing the cages should not be necessary for a basic cleaning and since they can be difficult to re-install, we recommend leaving them in place.

The uniqueness statement is interesting—it means that, no matter how you row reduce, you always get the same matrix in reduced row echelon form.

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Solving equations by elimination requires writing the variables x , y , z and the equals sign = over and over again, merely as placeholders: all that is changing in the equations is the coefficient numbers. We can make our life easier by extracting only the numbers, and putting them in a box:

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When we wrote our row operations above we used expressions like R 2 = R 2 − 2 × R 1 . Of course this does not mean that the second row is equal to the second row minus twice the first row. Instead it means that we are replacing the second row with the second row minus twice the first row. This kind of syntax is used frequently in computer programming when we want to change the value of a variable.

It will be very important to know where are the pivots of a matrix after row reducing; this is the reason for the following piece of terminology.

Regularly cleaning your bearings not only keeps you rolling smooth and fast on the track but it extends the life of your bearings, too! This guide includes a video and helpful step-by-step instructions to cover all of the basics of bearing cleaning.

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That is to say, the columns of the augmented matrix are exactly the column-vectors appearing in the equation. The column of constants in the equation, on the right of the “= ” sign, becomes the augmentation column of the matrix.

What has happened geometrically is that the original blue line has been replaced with the new blue line y = 1. We can think of the blue line as rotating, or pivoting, around the solution ( 1,1 ) . We used the pivot position in the matrix in order to make the blue line pivot like this. This is one possible explanation for the terminology “pivot”.

The cleaning solution you choose is very important, as you do not want to leave any residue that might collect debris or interfere with the lubrication. There are premade solutions made for bearings but rubbing alcohol works, too, just make sure it is 90% ABV or higher. Rubbing alcohol also does wonders to remove any film left from soap or other cleaners, just in case.

In the previous subsection we saw how to translate a system of linear equations into an augmented matrix. We want to find an algorithm for “solving” such an augmented matrix. First we must decide what it means for an augmented matrix to be “solved”.

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We can visualize this system as a pair of lines in R 2 (red and blue, respectively, in the picture below) that intersect at the point ( 1,1 ) . If we subtract the first equation from the second, we obtain the equation 2 y = 2, or y = 1. This results in the system of equations:

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Replace the shields that you removed earlier by snapping them into place. Check each bearing's spin after you do to make sure they are flush and spinning freely. Then, re-install your bearings into your wheels - again, a bearing tool is extremely useful here - and finally, your wheels on to your skates.

When deciding if an augmented matrix is in (reduced) row echelon form, there is nothing special about the augmented column(s). Just ignore the vertical line.

If an augmented matrix is in reduced row echelon form, the corresponding linear system is viewed as solved. We will see below why this is the case, and we will show that any matrix can be put into reduced row echelon form using only row operations.

We will give an algorithm, called row reduction or Gaussian elimination, which demonstrates that every matrix is row equivalent to at least one matrix in reduced row echelon form.

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It is essential that your bearings be completely dry before applying lubricant. If using a premade cleaner, follow their given instructions. Rubbing alcohol will evaporate on its own without residue, which is why it is recommended. In any case, you could always use a blow dryer to speed up the process. Once you are confident that your bearings are totally dry, apply 2 drops of your bearing lubricant to each bearing and give them a good spin.

We will solve systems of linear equations algebraically using the elimination method. In other words, we will combine the equations in various ways to try to eliminate as many variables as possible from each equation. There are three valid operations we can perform on our system of equations:

Place the bearings in the container with the cleaning solution and secure the lid. Agitate for 30 seconds to make sure that the solution reaches every nook and cranny in each bearing. Afterwards, let them sit undisturbed for 3-5 minutes while the sediment and other build-up settles to the bottom. While you wait, prepare an area where the bearings can be dried and lubricated.

First, remove all eight wheels from your skates and then carefully extract all sixteen bearings from those wheels. Be sure to apply force only on the metal races of the bearings as the shields and cages are easily damaged. A bearing tool will greatly simplify this step and ensure that bearings are not damaged in the process.