Solve each system, if possible. { 2 x+y-z=7 x-y+z=2 x+y-3 z=2 .

Name: Problem 56, Chapter 6: College Algebra
Uploaded: 2020-07-11T01:00:25-08:00
Duration: 4 min 46 s
Channel: Vicki Stebbins
Description: Solve each system, if possible. { 2 x+y-z=7 x-y+z=2 x+y-3 z=2 .

step 1 All right, first let's label these. I'm going to call this one A, this one B, and this one C. An

Solve each system, if possible. { 2 x+y-z=7 x-y+z=2...

Key Concepts

System of Linear Equations

This concept involves a group of equations that all share the same set of variables. The task is to find values for these variables that satisfy each equation simultaneously. The system can have a unique solution, infinitely many solutions, or no solution at all, depending on whether the equations are independent or redundant.

Gaussian Elimination

Gaussian elimination is a systematic method used to solve systems of linear equations. It involves applying row operations to convert the system's augmented matrix into a row-echelon or reduced row-echelon form, making it easier to solve for the variables through back-substitution.

Consistency and Uniqueness

These are key ideas when working with systems of equations. A system is consistent if it has at least one solution and inconsistent if no common solution exists. Additionally, a consistent system may have a unique solution if there are as many independent equations as variables, or infinitely many solutions if there are fewer independent equations than variables.

Substitution Method

The substitution method is another technique for solving linear systems, where one of the equations is solved for one variable in terms of the others. This expression is then substituted into the remaining equations, reducing the number of variables and making it simpler to find the solution.

Transcript

00:01 All right, first let's label these.

00:02 I'm going to call this one a, this one b, and this one c.

00:12 And we want to look to see where we can zero out variables.

00:16 Well, let's do a and b and zero out the y.

00:23 So a then is 2x plus y minus z equals 7.

00:37 X minus y plus z equals 2 and look i think we're going to get really lucky here so these zero out these zero out so i'm left with 3x equals 9 so that gives me x equals 3 well that's kind of cool made that easy all right and now let us pair which one will dub zero out uh let's let's get rid of, yeah, let's get rid of so many decisions.

01:27 Let's go with b, which is x minus y plus z equals 2.

01:37 And let's get rid of c.

01:40 Actually, now what we can do, now that i think about it, plus y equals 3z.

01:47 Okay.

01:48 Now these are going to zero out.

01:52 And then we have this equation d, which is 2x plus 4 z equals 4.

02:07 You don't know why i have an equal sign there.

02:11 And then guess what we can do? we can put that into there.

02:22 And so i'll have 2 times 3 plus 4 z equals 4.

02:29 So i have 6 plus 4 z equals 4.

02:40 I feel like i did something wrong.

02:43 Hold on a minute...