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

Name: Problem 25, Chapter 10: Precalculus
Uploaded: 2020-07-31T22:04:59-08:00
Duration: 6 min 19 s
Channel: Bobby Barnes
Description: Find all solutions of each system. { 2 x+3 y+2 z =5 x+4 y-3 z =1 .

step 1 They want us to solve this system of linear equations here. So now first notice that we only hav

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

Key Concepts

Elimination Method

The elimination method is a technique used to solve systems of linear equations by systematically canceling out variables. This is done by adding or subtracting equations to reduce the number of variables, eventually simplifying the system so that the solution can be expressed in terms of free parameters if necessary.

Parametric Representation

This concept refers to the method of expressing the general solution of a system in terms of one or more independent parameters (free variables). Parametric representation provides a clear and concise way to describe an infinite set of solutions by linking dependent variables to free parameters.

Underdetermined Systems

An underdetermined system has fewer independent equations than unknown variables, which generally implies that there are infinitely many solutions. Understanding these systems involves recognizing that some variables cannot be uniquely determined and must be expressed in terms of one or more free parameters.

Free Variables

Free variables are those that are not constrained by the system's equations, typically appearing in underdetermined systems. These variables can take on any value, and the other variables in the system are expressed in terms of these free variables, leading to a parameterized description of all solutions.

Systems of Linear Equations

This concept involves a set of equations where each equation is linear in its variables. The goal is to find values for the variables that satisfy all of the equations simultaneously. These systems serve as the foundation for algebraic problem-solving and have applications across science, engineering, and economics.

Transcript

00:01 They want us to solve this system of linear equations here.

00:05 So now first notice that we only have two equations, but we have three variables.

00:09 So that means we're either going to have a system with infinitely many solutions, or we're going to have an inconsistent system, so no solution at all.

00:19 So let's go ahead and see which one of those we end up with.

00:23 So now we would want to first try to clear that x out.

00:29 So normally we would go down to do this, but i'm going to clear it going up just because i don't want to have to do with fractions.

00:36 So what i'm going to do is i'm going to multiply row 2 by negative 2, and then add that result to row 1.

00:48 So 1 times negative 2 is negative 2, add that to 2, so the x is cancel.

00:54 And then we'd have 4 times negative 2, 8, negative 8, add that to 3.

01:02 That would give us negative 5y, and then negative.

01:05 3 times negative 2 is 6 add that to 2 that gives us 8 z and then 1 times negative 2 negative 2 add that to 5 we end up with 3 and then the second equation we didn't do anything with so that's just x plus 4 y minus 3 z is equal to 1 right so let's get rid of that 0 now all right, so now notice that if we were to just switch rows 1 and row 2, so i'll just go ahead and do that.

01:38 Because remember, it doesn't really matter which one is on top of the other.

01:42 So if we were to write it like this, now notice it's in that same kind of lower or upper triangle matrix that we've seen before.

01:55 And notice that z doesn't have its own pivot column, and because of that, z is going to be our free variable.

02:02 Z is our free variable, which means we want to solve both of these equations in terms of z.

02:12 And then at the end, we'll just say z is some arbitrary variable.

02:17 So that second equation, so negative 5y plus 8c is equal to 3.

02:24 So we'll subtract 8x over.

02:28 Now give us 3 minus 8c, divide you side by negative 5.

02:32 Side gives us y is equal to 3 minus 8c, both.

02:36 Over negative 5 and then we could distribute that negative and doing that would give us or actually let's just distribute the negative 5 altogether so that would just give us 8 5s z minus 3 5ths and that'll just kind of save us a little bit of time when we go to plug it in for solving for x so now we had x plus 4 y minus 3 z is equal to 1 and now we we want to plug in y, which we found to these.

03:13 Actually, i need a little bit more space to plug that in.

03:17 So let's erase that.

03:21 And y is 8 fifths z minus three fifths...

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