2 linear equations in 2 unknowns:a11 x a12 y = b1 a21 x a22 y = b2

goal: find all x, y making both of these equations true at once. Since the solution set to each single equation is a line one geometric interpretation is that you are looking for the intersection of two lines.

Exercise 3: Consider the system of two equations E1, E2:E1 5 x 3 y = 1 E2 x 2 y = 8

3a) Sketch the solution set in 2, as the point of intersection between two lines.

3b) Use the method of substitution to solve the system above, and verify that the algebraic solution yields the point you estimated geometrically in part a.

3c) What are the possible geometric solution sets to 1, 2, 3, 4 or any number of linear equations in two unknowns?

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Math 2270-002 Tues Aug 21 1.1 Systems of linear equations and Gaussian elimination 1.2 Row reduction, echelon form, and reduced row echelon form

Announcements:

Warm-up Exercise:

Our Canvaspage is liveI've posted HW 1 on publicpage on CANVAS startHWChapter 1 of text is on CANVAS soon

Quiz 1 tomorrow on Gaussianelimination

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b Solve the system in 1a algebraicallySketch

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This is the system we solved at the end of class on Monday:

E1 5 x 3 y = 1 E2 x 2 y = 8

The method of substituion that we used then does not work well for systems of equations with lots of variables and lots of equations. What does work well is Gaussian elimination, which we now explain andillustrate.

1a) Use the following three "elementary equation operations" to systematically reduce the system E1, E2 to an equivalent system (i.e. one that has the same solution set), but of the form

1 x 0 y = c1 0 x 1 y = c2

(so that the solution is x = c1, y = c2). Make sketches of the intersecting lines, at each stage.

The three types of elementary equation operation that we use are listed below. Can you explain why the solution set to the modified system is the same as the solution set before you make the modification? interchange the order of the equations multiply one of the equations by a non-zero constant replace an equation with its sum with a multiple of a different equation.

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1b) Look at our work in 1a. Notice that we could have saved a lot of writing by doing this computation "synthetically", i.e. by just keeping track of the coefficients in front of the variables and right-side values: We represent the system

E1 5 x 3 y = 1 E2 x 2 y = 8

with the augmented matrix

5 3

1 2

1

8 .

Using R1, R2 as symbols for the rows, redo the operations from part (a). Notice that when you operate synthetically the "elementary equation operations" correspond to "elementary row operations": interchange two rows multiply a row by a non-zero number replace a row by its sum with a multiple of another row.

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Solutions to linear equations in 3 unknowns:

What is the geometric question we're answering in these cases?

Exercise 1) Consider the system x 2 y z = 4

3 x 8 y 7 z = 202 x 7 y 9 z = 23 .

Use elementary equation operations (or if you prefer, elementary row operations in the synthetic version) to find the solution set to this system. The systematic algorithm we use is called Gaussian elimination. (Check Wikipedia about Gauss - he was amazing!)

Hint: The solution set is a single point, x, y, z = 5, 2, 3 .

2 I 43 8 7 20

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212,1123

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Exercise 2) There are other possibilities. In the two systems below we kept all of the coeffients the same as in Exercise 1, except for a33 , and we changed the right side in the third equation, for 2a. Work out what happens in each case.

2a)x 2 y z = 4

3 x 8 y 7 z = 202 x 7 y 8 z = 20 .

2b)x 2 y z = 4

3 x 8 y 7 z = 202 x 7 y 8 z = 23 .

2c) What are the possible solution sets (and geometric configurations) for 1, 2, 3, 4,... equations in 3 unknowns?

Summary of the systematic method known as Gaussian elimination for solving systems of linear equations.

We write the linear system (LS) of m equations for the vector x = x1, x2,...xn of the n unknowns asa11 x1 a12 x2 ... a1 n xn = b1 a21 x1 a22 x2 ... a2 n xn = b2

: : :am1 x1 am2 x2 ... am n xn = bm

The matrix that we get by adjoining (augmenting) the right-side b-vector to the coefficient matrix A = ai j is called the augmented matrix A b :

a11 a12 a13 ... a1 n

a21 a22 a23 ... a2 n

: :

am1 am2 am3 ... am n

b1

b2

:

bm

Our goal is to find all the solution vectors x to the system - i.e. the solution set .

There are three types of elementary equation operations that don't change the solution set to the linear system. They are interchange two of equations multiply one of the equations by a non-zero constant replace an equation with its sum with a multiple of a different equation.

And that when working with the augmented matrix A b these correspond to the three types ofelementary row operations: interchange ("swap") two rows multiply one of the rows by a non-zero constant replace a row by its sum with a multiple of a different row.