Students are buying books for the new semester. Eddie buys the environmental statistics book and

Students are buying books for the new semester. Eddie buys...

Key Concepts

System of Linear Equations

This concept involves setting up and solving multiple linear equations simultaneously to find the values of unknowns. In many real-world scenarios, such as determining individual prices from total costs, a system of linear equations can model the relationships between different quantities, and solving the system yields the specific values for each unknown.

Variable Assignment

Assigning variables to unknown quantities is a fundamental step in algebraic modeling. It involves representing each unknown (such as the price of a book) with a symbol, making it easier to express relationships and constraints described in a word problem as mathematical equations.

Elimination and Substitution Methods

These are standard techniques used to solve systems of equations. The elimination method involves combining equations to cancel out one variable, while the substitution method involves solving one equation for a variable and then replacing that variable in another equation. Both methods are crucial for efficiently finding the values of unknowns in a system.

Translating Word Problems into Equations

This key concept refers to the ability to convert the narrative details of a word problem into a coherent set of mathematical equations. It requires identifying the quantities involved, understanding their relationship as described in the problem, and then formulating equations that accurately represent the scenario.

Transcript

00:01 We can use some linear algebra with some matrices and some gaussian elimination to solve this.

00:08 So first i'm going to start out with just a quick three variables.

00:15 So we're going to use e for environmental statistics.

00:19 We're going to use s for set theory.

00:22 We're going to use p for educational psychology.

00:26 And using those three variables, we can set up now three equations.

00:30 So we have one environmental statistics book plus a set theory book, and that was going to equal $178.

00:41 We have $319 being equal to two environmental statistics books, all right, one set theory book, and one educational psychology book.

00:58 And then we are going to have one educational psychology book plus one set theory book, and that is going to be 147.

01:11 So i can turn this into some matrices and just go through some gaussian elimination with this.

01:21 So we're going to have one e, right, one e, one s, and then zero p, and that is going to give us 178.

01:31 And our second row will be 2, 1, 1, and then 319.

01:36 Right, i can set this up just using the coefficients of those variables, as long as i keep my variables all in line.

01:45 Then we'll have 0 -1 -1, and that will be 147.

01:53 So going just from matrix to matrix here, as we go through this elimination process, i'm just going to label the step that we take.

02:02 So if we take negative 2 of row 1 and add it to row 2 to get a new row 2, we will have rows 1 and 3 looking the same.

02:20 But this middle row will have negative 2 times 1 plus 2 will be 0.

02:30 Negative 1 times 2, yeah, sorry, negative 2 times 1 plus 1 will give us negative 1, and 0 times negative 2 is 0 plus 1 is 1.

02:41 But that trickier one on the end is that 178 times negative 2 plus the 319, all right, will give us negative 37.

02:54 And our next elimination, remember, we want to get.

02:58 It to be in a reduced row echelon form where we only have ones on the diagonal and zeros everywhere else.

03:05 We can now add row two to row three.

03:16 Okay, so rows one and two will remain the same.

03:27 And this is the row that's just going to change.

03:35 Okay, so we have negative one plus one is zero.

03:38 1 plus 1 is 2, and negative 37 plus 147 will be 110.

03:47 So now i'm going to do a double switch here, okay, because i'm going to do just multiply two different rows by their own scalar scalar multiples.

03:57 So i'm going to multiply row 2 by negative 1, and i'm going to multiply row 3 by 1 half.

04:08 Okay, so row one will stay the same...

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