Find all solutions of each system. { -2 x+2 y-z=0 3 x-4 y+z=1 5 x-8 y+z=4 .

Name: Problem 20, Chapter 10: Precalculus
Uploaded: 2020-07-31T22:03:31-08:00
Duration: 4 min 51 s
Channel: Bobby Barnes
Description: Find all solutions of each system. { -2 x+2 y-z=0 3 x-4 y+z=1 5 x-8 y+z=4 .

Find all solutions of each system. { -2 x+2 y-z=0 ...

Key Concepts

Systems of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. Understanding the structure of these systems is fundamental to solving problems in algebra and is applicable in various fields such as engineering, economics, and physics.

Elimination Method

The elimination method is a technique used to solve systems of linear equations by combining equations to cancel out one or more variables. This method simplifies the system step-by-step until it can be solved by back substitution, making it a powerful tool for both small and large systems of equations.

Consistency and Inconsistency

The concepts of consistency and inconsistency pertain to whether a system of equations has solutions. A system is consistent if there is at least one set of values that satisfies all equations, and inconsistent if no such solution exists. Detecting contradictory equations during the solution process is key to determining the nature of the system.

Transcript

00:01 They want us to find all of these solutions for this set of linear equations here.

00:06 So now what we would normally do is we would try to turn this into an upper triangular matrix and then go from there to get everything to cancel out.

00:17 But notice that if we try to cancel out the x's, then we end up having to use fractions.

00:24 And i don't know about shaw, but i really don't like half fain to use fractions.

00:29 So what i'm going to do is, instead of doing an upper triangular matrix, i'm going to make a lower triangle, triangular matrix.

00:36 So what we're going to do, instead of using x, i'm going to start over here with z and then work my way up.

00:44 So we're going to start with this z here, and we want to cancel everything above it.

00:51 So let's see what would be good to start with.

00:56 So that means, well, to get this z to cancel out with the one above it, we would just multiply it by 1 or negative 1.

01:04 So it would be negative row 3 plus row 2.

01:10 And for that top row, to get that to cancel out, we would just add it.

01:15 So it would be r3 plus row 1, like that.

01:21 So let's go ahead and write this out.

01:24 So the first row, we're adding row 3 to it.

01:32 And doing that will give us 3x, because we're doing 5x plus negative 2x and then negative 8y plus 2y so that would be negative 6y and then z plus negative z that's just 0 and then 4 plus 0 is 4.

01:58 Now for the second one so we do negative 1 times 5 negative 5 plus 3 that would give us negative 2 so we have negative 2x and then negative 8 times 1 that gives us 8 plus negative 4 that gives us plus 4y, and then negative z plus z would just give us 0.

02:22 And then negative 4 plus 1 would be negative 3.

02:28 And then we didn't change this last one.

02:31 So it's still 5x minus 8y plus z is equal to 4.

02:38 So now let's go ahead and clear these zeros out.

02:43 So now we want to clear this y here.

02:53 So we need to multiply 4 y by something.

02:57 So it looks like i'll still have to use fractions, which is unfortunate...