MATH 1046 Introduction to Systems of Linear Equations (Sections 2 MATH 1046 Introduction to Systems of Linear Equations (Sections 2.1 and 2.2) Alex Karassev

Example: Error in scale Alex is using a scale that is known to have a constant error. A can of soup and a can of tuna are placed on this scale, and it reads 720 grams. Four identical cans of soup and three identical cans of tuna are placed on an accurate scale, and a weight of 2.4 kg is recorded. If two cans of tuna weigh 540 grams on this bad scale, what is the amount of error in the scale and what is the correct weight of each type of can?

Mathematical Model 4 cans of soup and 3 cans of tuna on good scale equals 2.4 kg A can of soup and a can of tuna on bad scale equals 720 g 2 cans of tuna on bad scale equals 540 g Let x, y, and z be the weights of a can of soup, a can of tuna, and the error of the scale in grams, respectively. Then we obtain the following system of linear equations:

Example: Logging A logging company has a contract to provide 1000 m³ of pine, 800 m³ of spruce, and 600 m³ of fir logs per month. There are three regions available for logging. The following table gives the species mix, and timber density for each region. How many hectares should one log in each operating region listed above to deliver exactly the required volume of logs? Region Volume/hectare % Pine % Spruce % Fir West 330 m³ 70 % 20 % 10 % North 390 m³ 60 % 30 % East 290 m³ 5 % 75 %

Mathematical Model Region Volume/hectare % Pine % Spruce % Fir West 330 m³ 70 % 20 % 10 % North 390 m³ 60 % 30 % East 290 m³ 5 % 75 % We need: 1000 m³ pine, 800 m³ spruce, and 600 m³ fir Let w, n, e be the number of hectares logged in West, North, and East regions respectively. Then:

Modern applications of linear systems Linear systems with hundreds (or more!) unknowns: Applications in economics (linear optimization (programming), transportation problem, linear models) Computer graphics Optimization Traffic flow

How to solve linear systems? Straightforward approach: express one of the variables form one of the equations and substitute in the rest of equations Continue until you get to one equation with one variable

Error in scale: solution From last equation: z = 540 – 2y Substitute in the first, we get x+ y + (540 – 2y) = 720 So, we get two equations: x – y = 180 4x + 3y = 2400 Now from the first equation x = 180 + y and therefore 4(180 + y) + 3y = 2400 Thus, 7 y = 1680, so y = 240 Now x = 180 + 240 = 420 and z = 540 – 2y = 60

Problem with straightforward approach It is not efficient (e.g. computationally) when the number of equations is large

What can we do with systems of equations? Example Divide all by 10 Divide 2nd by 2

What can we do with systems of equations? Example Subtract 2nd from 3rd Change the order of 1st and 2nd

What can we do with systems of equations? Example Multiply 1st by 7 and subtract from 2nd Add 3rd to 2nd

Matrix form Coefficient matrix Augmented matrix

Matrix form General 3x3 system

So, we can: Change the order of equations Multiply an equation by a nonzero number Add a multiple of one equation to another equation Note: all this operations are invertible, i.e. the system can be returned to its original form by applying the same types of operations

Elementary row operations on augmented matrix Switch two rows Multiply a row by a nonzero number Add a multiple of one row to another row Note: all this operations are invertible, i.e. the matrix can be returned to its original form by applying the same types of operations

Notations Switch 1st and 2nd rows: R1  R2 Multiply 1st row by a nonzero number: aR1 Add a multiple of 1st row to the 2nd row What are their inverses?

“Direct” and “Inverse” operations Switch 1st and 2nd rows: R1  R2 Multiply 1st row by a nonzero number: aR1 Add a multiple of 1st row to the 2nd row: R2 → R2 + bR1 Inverse Multiply 1st row by a nonzero number: (1/a)R1 Add a multiple of 1st row to the second row: R2 → R2 + (-b)R1

Goal of the operations If possible, reduce the augmented matrix to the form: Then x1=r1, x2=r2, x3=r3 is the (unique) solution

Linear systems and geometry Systems of linear equations can be used to find intersections of planes and lines Exercise Find the point at which the following planes intersect x+y+z=3, x+2y-z=3, 2x+3y+z=1