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

Systems of Linear Equations

A system of linear equations is a collection of one or more linear equations involving the same set of variables. The goal is to find the values of the variables that satisfy all the equations simultaneously, making it a fundamental representation of multiple constraints acting on the same quantities.

Consistency and Inconsistency

The concept of consistency in a system of equations refers to whether there is at least one solution that satisfies all equations simultaneously. If at least one solution exists, the system is considered consistent; if no solution exists due to conflicting equations, it is considered inconsistent.

Elimination Method

The elimination method is a technique used to solve systems of linear equations by strategically adding or subtracting equations to eliminate one or more of the variables. This stepwise reduction simplifies the system, making it easier to solve for the remaining variables.

Gaussian Elimination

Gaussian elimination is a systematic procedure for solving systems of linear equations. It involves representing the system as an augmented matrix, then performing row operations to reach row echelon form, and finally using back-substitution to determine the values of the variables.

Transcript

00:01 In this problem, we're going to look at how to do a system of equations with a set of three equations.

00:08 First thing we're going to need to look at is which one of our equations can be eliminated using simply two of the three equations.

00:18 Okay? so let's go ahead and finish writing these out first.

00:33 Now, if i look at this set of three equations, i'm going to choose two of them to work with by their selves.

00:41 First.

00:42 Now our my goal is i want to eliminate one of my variables so i want to see which two equations will allow me to very easily eliminate one single variable.

00:53 So let's choose these first equation and let's choose this second equation.

01:00 As we can see here 2x can be eliminated from 2x so let's rewrite it so it's very clear.

01:17 All right so this is going to get eliminated and we're left with 3y minus 2z equals negative 7.

01:26 Now we're going to repeat that process and we're going to try to eliminate x again, but this time using two different equations.

01:36 Okay, so let's just go ahead and use this equation here and the third equation.

01:42 All right.

01:43 So let's rewrite this.

01:50 Now, in order to eliminate x, we need to have the exact same coefficient.

01:55 With opposite signs.

01:57 We have the opposite signs, but we need the exact same coefficient.

02:02 So let's go ahead and multiply these two equations so that we get the exact same coefficient.

02:07 So multiply the top equation by three and the bottom equation by two.

02:12 All right? so when we rewrite that, we'll end up with negative 6x plus 6y plus 3z equals negative 21 and in the second equation we'll have positive 6x minus 8y minus 6c equals 14.

02:38 Now we can go ahead and eliminate x and here we'll end up with negative 2y minus 3z equals negative 7.

02:50 So now that we have two equations that have x eliminated, we can use these two equations to eliminate another variable.

03:01 So let's go ahead and stack these two equations together.

03:06 So 3y minus 2z equals negative 7 and minus 2y minus 3z equals negative 7.

03:16 Now we need to eliminate one of these variables.

03:19 So we're going to eliminate by making sure they have the exact same coefficients.

03:27 So let's go ahead and eliminate y.

03:31 So if we want to eliminate y, we're going to go ahead and multiply this top equation by 2.

03:38 And we're going to multiply the bottom equation by 3.

03:43 So that's going to give me two new equations of 6y minus 4z.

03:52 Equals negative 14 and negative 6y minus 9 z equals negative 21 combined these two equations the ys are eliminated negative 13 z is going to be equal to negative 35 last step let's go ahead and divide by negative 13 on both sides and we get that z is equal to a positive 35 over 13 so we have our first variable now we need to solve for our next variable in order to do that we're going to have to take this z and plug it into one of the equations that we solve for here so let's use the very top equation right so we have 6 y minus 4 z is equal to negative 14 now we know know based off of this variable here that z is equal to 35 over 13.

04:59 So we're going to write this as 6y minus 4 times 35 over 13 is equal to negative 14.

05:11 Now 4 times 35 over 13 is going to give us 140 over 13.

05:18 So 6y minus 140 over 13...

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