Solve for Three Unknowns Using Matrices and Row Operations

!±8± Solve for Three Unknowns Using Matrices and Row Operations

A system of linear equations can be solved by (1) preparing an Augmented Matrix, and then (2) using row operations to convert the Augmented Matrix into a triangular form. Once the Augmented Matrix has been converted to triangular form, the solution to every variable can often be read directly from the matrix with no further work.

This can be illustrated by solving for the variables x, y, and z in the following system of linear equations:

-4x - 3y - z = -12
-5x + 6y + 4z = -24
-3x - y - 6z = -22

First, arrange all three equations in standard form.

Ax + By + Cz = D

This has already been done in the statement of the problem:

-4x - 3y - z = -12
-5x + 6y + 4z = -24
-3x - y - 6z = -22

Prepare an Augmented Matrix of 3 rows and 4 columns.

Each row in the Augmented Matrix will represent one of the three equations.

The first number in each row is the coefficient for the x variable, the second number in each row is the coefficient for the y variable, the third number in each row is the coefficient for the z variable, and the last number will be the constant:

| -4 -3 -1:-12 |
| -5 +6 +4: -24 |
| -3 -1 -6:-22 |

THIS IS the GOAL: Convert the Augmented Matrix into a triangular form. Once this is done, we are finished.

The triangular form will look like this: a diagonal pattern of 1's with 0's everywhere else but the last column.

| 1 0 0:? |
| 0 1 0:? |
| 0 0 1:? |

Once we have converted the matrix into the triangular form, the? shown in row 1 will be the solution value of x, the? shown in row 2 will be the solution value of y, and the? shown in row 3 will be the solution value of z.

To convert the augmented matrix into a triangular matrix, we will perform various row operations, one at a time.

To convert the convert the Augmented Matrix to a triangular form, the following operations can be used:

1. switch two of the rows
2. multiply any row by a number which is not zero
3. Replace any row with the result of adding that row to another row.

| -4 -3 -1:-12 |
| -5 +6 +4: -24 |
| -3 -1 -6:-22 |

Note:

There are many calculations involved, which means there is a high probability of making errors in arithmetic. For this reason I recommend using a calculator.

| -4 -3 -1:-12 |
| -5 +6 +4: -24 |
| -3 -1 -6:-22 |

Multiply R 1 by -1/4

| 1 3/4 1/4: 3|
| -5 +6 +4: -24 |
| -3 -1 -6:-22 |

add 5 times the 1st row to the 2nd row

| 1 3/4 1/4: 3|
| 0 39/4 21/4: -9|
| -3 -1 -6:-22 |

add 3 times the 1st row to the 3rd row

| 1 3/4 1/4: 3|
| 0 39/4 21/4: -9|
| 0 5/4 -21/4:-13|

multiply the R 2 by 4/39

| 1 3/4 1/4: 3|
| 0 1 7/13: -12/13|
| 0 5/4 -21/4:-13|

add -5/4 times the 2nd row to the 3rd row

| 1 3/4 1/4: 3|
| 0 1 7/13: -12/13|
| 0 0 -77/13:-154/13|

multiply the R 3 by -13/77

| 1 3/4 1/4: 3|
| 0 1 7/13: -12/13|
| 0 0 1: 2|

add -7/13 times the 3rd row to the 2nd row

| 1 3/4 1/4: 3|
| 0 1 0: -2|
| 0 0 1: 2|

add -1/4 times the 3rd row to the 1st row

| 1 3/4 0: 5/2|
| 0 1 0: -2|
| 0 0 1: 2|

add -3/4 times the 2nd row to the 1st row

| 1 0 0: 4|
| 0 1 0: -2|
| 0 0 1: 2|

This is the final triangular form of the Augmented Matrix (the row echelon form):

| 1 0 0: 4|
| 0 1 0: -2|
| 0 0 1: 2|

The solution to the three simultaneous equations is:

From R 1, x = 4
From R 2, y = -2
From R 3, z = 2

The final answer is:

x = 4
y = -2
z = 2

This answer can be verified by substituting the numerical values of x, y and z into the original equations:

-4x - 3y - z = -12
-4*4 - 3*(-2) - 2 = -12, correct

-5x + 6y + 4z = -24
-5*4 + 6*(-2) + 4*2 = -24, correct

-3x - y - 6z = -22
-3*4 - (-2) - 6*2 = -22, correct

Since the numerical values of x, y, and z work in all three of the original equations, the solutions are correct.

Solve for Three Unknowns Using Matrices and Row Operations

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Teach Your Kids Arithmetic - Division Shortcuts

!±8± Teach Your Kids Arithmetic - Division Shortcuts

The famous Greek mathematician Pythagoras (you know the one with that theorem) said, "Numbers have a way of taking you by the hand and leading you down the path of reason." What Pythagoras was getting at, I think, is that numbers--by their very nature--permit us to do things which enable an understanding of the very universe and its intricate laws. Numbers have their roots in arithmetic, and a mastery of this field, particularly the operations of addition, subtraction, multiplication, and--yes that monster operation of division--will certainly pave a smooth road down that path of reason.

For most children, the tedious task of memorizing their number facts evokes long-winded yawns and puts pain-riddled expressions on their cherubic faces. However, rote tasks such as these (see my article Mastering Arithmetic and Singapore - What's the Connection?) are what insure that children have a chance to progress up the ladder in mathematics. Of the four arithmetic operations, division is the one that children find most uncomfortable as this is the one operation which produces "uneven" results in the form of remainders. Such remainders thrust children into the world of fractions, percents, and decimals (see my article on this topic) and we all know how painful those things called fractions can be.

Given the difficulties that division presents to children learning their arithmetic basics, it would be both expedient and practical to present an approach which would lead them down this path of reason. Of the myriad number tricks and math shortcuts that I have developed over the years, the method of division by ten-multiples is one I am quite fond of. Basically, this method hinges on understanding the basic multiplication facts. After all, mathematics is a building blocks discipline, and this method, by being dependent on having mastered multiplication, clearly shows this fact.

This method shows an effective way to divide a three digit number by a one digit number. Take the example 306/6. A ten-multiple of 6 is 60. What we are trying to do is see how many ten-multiples of 6 go into 306 without going over. To do this effectively, we need to know our 6-times table. We know that 6x5 = 30 and 6x6 = 36. The respective ten-multiples of 30 and 36 are 300 and 360. Since 300 is less than 306, we know that 306 divided by 6 is going to be "50-something." After we divide out the 300, we are left with 6, which gives another 1 as part of the quotient. Thus 306/6 is 51.

Play with this method a little and teach it to your kids. The more exposure they have to arithmetic techniques, the more they will come to master mathematics. Who knows, they might even become whiz kids.

See more at my cool math site Arithmetic Magic and Fractions Troubleshooter.

Teach Your Kids Arithmetic - Division Shortcuts

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