High School: Algebra
Reasoning with Equations and Inequalities HSA-REI.C.8
8. Represent a system of linear equations as a single matrix equation in a vector variable.
Students should understand that a system of equations can be expressed as a matrix of coefficients multiplied by a vector matrix of variables. Sometimes, it's even better to do it that way.
Your students will probably curse you for stressing matrices time and time again, but just reassure them that it's for their own good. In order to work with matrices, your students should already understand what they are and how to perform various functions with them. The same goes for linear equations and systems of linear equations.
Students should know that a matrix equation takes the form AX = B, where A represents the coefficient of our variables, X represents our variables, and B represents the output to our equations.
In order to turn linear equations into matrices, we have to rearrange them all into the same format. Among the preferred formats are ax + by + cz = d. The linear equations y = 2x + 5 and y = 3x – 2 become -2x + y = 5 and -3x + y = -2. Painless, no?
Then we take each term in the equations and split them up into their proper matrix. Taking the equations from above, we'll have the A matrix (the one with all the coefficients) equaling
Since our only variables are x and y (and we put the coefficients into the A matrix in that order), our X matrix becomes:
The final matrix, B, is what's on the other side of our equal signs. That means it turns into:
In matrix form, it looks like this:
This standard only indicates that students should know how to represent systems of linear equations in matrix equation form, but not why doing so is useful (aside from organizational purposes). That is covered in the next standard.