Solve each system of equations. If the system has no solution, state that it is inconsistent. {

Solve each system of equations. If the system has no...

Key Concepts

System of Linear Equations

A system of linear equations is a set of two or more linear equations involving the same set of variables. The goal is to find the variable values that satisfy all equations at once, forming the solution to the system.

Consistency and Inconsistency

This concept refers to whether a system of equations has a solution. A consistent system has at least one solution (unique or infinitely many), while an inconsistent system has no solution, which is critical in analyzing the feasibility of the system.

Gaussian Elimination

Gaussian elimination is a systematic method for solving systems of linear equations. It involves using row operations to transform the system's augmented matrix into an upper triangular or row echelon form, from which the solutions can be determined via back-substitution.

Substitution and Elimination Methods

Both are classical techniques for solving systems of equations. The substitution method involves solving one equation for a variable and substituting it into another, while the elimination method focuses on adding or subtracting equations to remove a variable, simplifying the process of finding the remaining unknowns.

Transcript

00:02 So in this problem, we're given three equations with three n nodes and asked to solve them.

00:07 So we're given x plus 4y minus 3z equals negative 8.

00:19 3x minus y plus 3 z equals 12.

00:27 And x plus y plus 6 z equals 1 i'm going to draw a line across here just to keep our work a little neat call this formula 1 this formula 2 and this formula equation 3 okay so it's very easy to see that if i add 1 and 2 together then the z will drop out and i'll just be left with an equation in x and y so let's do that let's do 1 plus 2.

01:05 So x plus 3x is 4x.

01:09 4y minus y, that's plus 3y.

01:12 The z's cancel, and minus 8 plus 12 is 4.

01:18 Okay.

01:20 Then let's take equation 1 and equation 3, and notice that if i multiply equation 1 by 2, 2 times equation 1, and then add equation 3 to that, the z will drop out.

01:39 So 2 times equation 1.

01:40 That's 2x plus 8y minus 6 z equals minus 16.

01:50 And equation 3x plus y plus 6 z equals 1.

01:57 Add these together.

02:00 3x plus 9y.

02:04 The z's cancel out.

02:08 Is minus 15.

02:11 Okay, i'm going to call this equation 4 and this equation 5.

02:17 Because now i can see that between equation 4 and 5, i have two equations with two unknowns.

02:23 So i can solve them.

02:26 So let's take minus 3 times equation 4.

02:29 That gives me minus 12x minus 9y is equal to negative 12.