Embedded Files
Putting your calculator to work to check your algebra
Putting your calculator to work to check your algebra
We have been trying to get our system of equations into a form which shows that either it is inconsistent, or that it has infinitely many solutions along a line.
One particular form of a system of equations is called reduced row echelon form. We don't really need to know much about how what this means. The important thing is that this system has the same solutions as the original system.
An example of the coefficients of a system of equations put into reduced row echelon form (RREF) is below.
The grid of coefficients is called a matrix.
A system of equations
x + 2y + 3z = 4
5x + 6y + 7z = 8
9x + 10y + 11z = 12
Some graphics calculators don't do this process. That's okay! This process is only useful for checking your the algebraic working that you must show for Merit and Excellence.
The equations are entered in the RUN-MAT (standard calculator) part of a Casio graphics calculator.
Press (F1) for [>MAT], and press (EXE) to make the matrix dimension "m=3" and "n=4" for 3×4, that is 3 rows and 4 columns.
Enter the numbers in the 12 cells in the usual way, then press (EXIT) twice.
Press (OPTN) (F2) for the [MAT] - matrices commands.
Press (F6) for more options [>].
Press (F5) for RREF.
Press (F6) to go back, and (F1) for [Mat].
Press (SHIFT) then (2) for Mat.
Press (ALPHA) and then (X,θ,T) for "A".
Calculator should show "Rref Mat A".
Press (EXE). The RREF form of the system of equations appears - it has replaced the matrix A.
In the example above, we can see that the system is consistent, but does not have a unique solution.
If z = t some real number, then
y + 2z = 3 means y = 3 - 2t
x - z = -2 means x = t - 2
The points on the line are in the form
(x,y,z) = (t - 2, 3 - 2t, t ) where t is any real number.
You would need to show how to arrive at this system of equations algebraically.
Step-by-step algebraic steps on a calculator
Step-by-step algebraic steps on a calculator
Your calculator can do simple operations on the form of a matrix of coefficients. When the matrix is onscreen, choose (F1) for [R-OP], which stands for row operations.
It can only do these three things.
(F1) [SWAP] - change the position of two rows.
(F2) [XRw] - replace a row by a multiple of itself. Calculator needs k (the multiple) and m (which row).
(F3) [XRw+] - an arcane name for sure! This adds a multiple of a row to another row. In particular, k times row-m gets added to row-n.
(F4) [Rw+] - is like the above, but with k = 1.
The numbers in the system of equations along the way give the algebra. This is shown in the example below.
Example
Example
In the system of equations above, we could eliminate x from equations 1 and 2 by adding (-5) lots of row 1 to row 2.
Calculator: [R-OP][XRw+] with k=-5, m=1 and n=2.
Similarly, we could eliminate x from equations 1 and 3 by adding (-9) lots of row 1 to row 3.
Calculator: [R-OP][XRw+] with k=-9, m=1 and n=9.
The new row 2 looks a bit messy, we could make it nicer by multiplying by (-1/4).
[R-OP][XRw] with k=-0.25 and m=2.
To eliminate y from the new equations 2 and 3 by adding 8 lots of row 2 to row 3.
[R-OP][XRw+] with k=8, m=2 and n=3.
This yields a row of zeroes; the system of equations is consistent, but does not have a unique solution.
At each stage, write down the algebra of the new equation formed.
(eq4) -4x -8y = -12 from (eq2) - 5(eq1)
(eq5) -8x - 16y = -24 from (eq3) - 9(eq1)
(eq6) x + 2y = 3 from -0.25(eq4)
0 = 0 from (eq5) + 8(eq6)
This is a quick and dirty introduction to matrices. There's a lot more to know, and a lot more you could learn here, about matrix determinants, multiplication, inverses and more. There isn't really the time.
You don't need to be able to do anything with RREF or row operations on a matrix to get Excellence in this standard.
The following website shows the working in solving systems of three equations with three unknowns.
Google Sites
Report abuse