Solve each system, if possible. If a system is inconsistent...

Key Concepts

Systems of Linear Equations

Systems of linear equations consist of multiple linear equations involving the same variables. The goal is to find the common values for these variables that satisfy every equation in the system. This concept is fundamental in algebra and underpins methods used in various fields including engineering, economics, and science.

Consistency and Inconsistency

A system of equations is considered consistent if there is at least one set of values that satisfies all the equations, and inconsistent if no such set exists. Recognizing consistency is critical because it determines whether a valid solution exists, affecting how the problem is approached and solved.

Dependent vs. Independent Systems

Dependent systems are those in which one or more equations can be derived from the others, leading to infinitely many solutions. In contrast, independent systems yield a unique solution. Identifying dependency is essential to understand the nature of the solution set and to decide if a system is simply a re-expression of the same constraint.

Elimination and Substitution Methods

These are common techniques for solving systems of equations. The substitution method involves solving one equation for one variable and substituting this expression into the other equations, while the elimination method focuses on adding or subtracting equations to cancel out a variable. Mastery of these methods provides a foundation for handling more complex systems.

Gaussian Elimination and Row Reduction

Gaussian elimination is a systematic method used to simplify systems of linear equations by transforming the corresponding augmented matrix into an upper triangular form or row-echelon form. This process not only simplifies the solving process via back-substitution but also helps in identifying whether the system is consistent, inconsistent, or dependent.

Transcript

00:01 So question 40, question 40 from lesson 12 .2, solve each system if possible, if a system is inconsistent or if the equations are dependent, state this.

00:18 Remember, inconsistent is no solution dependent infinitely many solutions.

00:25 So number 40, r minus s plus t equals 4, r plus 2s minus t equals negative 1, and r plus s minus 3t equals negative 2.

00:44 I see that equation 1 and equation 2 have additive inverses the t variable.

00:51 So that's the first thing i'm going to do.

00:53 I'm going to add equation one and equation two together and eliminate the t.

01:00 So r plus r is 2r, negative s plus 2s is s, and a positive t and a negative t equals 0, 4 minus 1 equals 3.

01:16 So now i'm going to look at equations 2 and 3.

01:21 If i multiply equation 2 by a negative 3, i can get additive inverses on the t variable again.

01:37 So negative 3 times r is negative 3r.

01:41 Negative 3 times 2s is negative 6s.

01:47 Negative 3 times t is plus 3t and negative 3 times negative 1 is a positive 3.

01:56 And i'm going to add that to equation number three, which is r plus s minus 3t equals negative 2.

02:09 Negative 3r and r equal negative 2 r.

02:13 Negative 6 and s equal a negative 5s.

02:20 The t's cancel out.

02:22 3 minus 2 is 1...

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