Solve Problems 5-26 using elimination by addition. x-y+z=1 2 x+y+z=6 7 x-y+5 z=15

Name: Problem 15, Chapter 8: Precalculus: Graphs and Models
Uploaded: 2020-06-23T21:43:43-08:00
Duration: 8 min 22 s
Channel: Cinsy Krehbiel
Description: Solve Problems 5-26 using elimination by addition. x-y+z=1 2 x+y+z=6 7 x-y+5 z=15

Solve Problems 5-26 using elimination by addition. ...

Key Concepts

System of Linear Equations

A system of linear equations consists of multiple linear equations that involve the same set of variables. The objective is to find a single set of values for the variables that satisfies every equation simultaneously, which can correspond to a unique solution, infinitely many solutions, or no solution at all, depending on the relationships among the equations.

Elimination Method (Addition Method)

The elimination method is a systematic procedure used to solve systems of linear equations by adding or subtracting equations to eliminate one or more variables. By strategically aligning and combining the equations, one reduces the system to fewer variables, ultimately simplifying it to a point where the remaining variables can be solved directly, and then back-substituted to find the full solution.

Consistency and Uniqueness

Understanding whether a system is consistent or inconsistent is key when solving equations. A consistent system has at least one solution, with a unique solution arising when the system’s equations are linearly independent. In contrast, if the equations are dependent or contradictory, the system may yield infinitely many solutions or be inconsistent, respectively.

Transcript

00:02 We have a system of three equations and three unknowns, and we want to solve using the elimination method.

00:09 So as we look at this, maybe a good thing to try would be to take a and b to eliminate the y.

00:18 So if we add a and b together, that y should vanish.

00:23 So let's see what happens here.

00:24 A is x minus y plus z equals 1, and b is 2x.

00:33 Plus y plus z equals six.

00:38 If we add those together, we get 3x plus 2z equals 1 plus 6 is 7.

00:49 And we'll call that d for now.

00:52 And we can also add equations b and c together.

00:56 We'll do the same thing.

00:57 So if we have b is 2x plus y plus z equals 6, and we have that c is 7x minus y plus 5 z equals 15.

01:15 If we add these two together, we get that 9x plus 6z equals 21.

01:26 So that's e.

01:28 Now, maybe you can see by inspection that d and e are essentially the same equate.

01:36 But if it's not obvious, let's say we wanted to eliminate the xes between the two of them.

01:43 So we're going to take, we'll go ahead and take equation d and multiply it by negative 3.

01:52 So we get negative 9x minus 6 z equals negative 21.

01:59 And when we add those together, we get that 0 equals 0, which is true.

02:05 And that tells us we might have infinitely many solutions.

02:11 So let's take a look at some other parts of it.

02:14 How about if we try to, let's assume we're going to parameterize.

02:22 Let's look at it that way.

02:23 So we'll say, maybe we're going to go ahead and say, y equals some real number p, right, where p is real.

02:34 So we're going to want to use our system of equations to either eliminate the x or the z.

02:39 So we can express the one that's left behind in terms of this p.

02:44 So how about let's eliminate the z using a and b.

02:50 So we'll take a equals x minus y plus z equals 1.

02:57 And if we want the z to be eliminated, we're going to have to multiply b by negative 1.

03:02 So that will be negative 2x minus y minus z.

03:09 Equals negative 6.

03:12 When we add those two together, x minus x is negative x.

03:17 Negative y and negative y is negative 2y equals negative 5...

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