Solve each system of linear equations. 2 x-3 y+4 z=-3 -x+y+2 z=1 5 x-2 y-3 z=7

Name: Problem 7, Chapter 8: Precalculus
Uploaded: 2020-06-15T15:29:03-08:00
Duration: 4 min 48 s
Channel: Ashley Volpe
Description: Solve each system of linear equations. 2 x-3 y+4 z=-3 -x+y+2 z=1 5 x-2 y-3 z=7

step 1 For this equation, I'm going to take my first and second equations. My first equation is 2x, 2x

Solve each system of linear equations. 2 x-3 y+4 z=-3 ...

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Key Concepts

Matrix Representation

Representing a system of linear equations in matrix form, particularly as an augmented matrix, helps in organizing the coefficients and constants. This representation is essential for applying algorithmic approaches such as Gaussian elimination or other matrix-based solution methods.

System of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of variables. The objective is to find the values of these variables that satisfy all of the equations simultaneously. Such systems are foundational in algebra and appear frequently in various applied mathematics contexts.

Gaussian Elimination

Gaussian elimination is a method used to solve systems of linear equations by systematically eliminating variables. This is done by performing row operations on the augmented matrix of the system to reach an upper triangular or row-echelon form, after which back substitution can be used to obtain the solution.

Solution Consistency and Uniqueness

In solving a system of linear equations, it is important to determine whether the system has a unique solution, infinitely many solutions, or no solution at all. This assessment is typically made by examining the rank of the coefficient matrix relative to that of the augmented matrix, which reflects the consistency of the system.

Transcript

00:01 For this equation, i'm going to take my first and second equations.

00:06 My first equation is 2x, 2x minus 3y plus 4z equals negative 3.

00:20 I'm going to combine that with my second equation, which is negative x plus y plus 2z, equals 1.

00:29 Now, since my coefficients don't match up, i'm going to multiply this second equation then by 2.

00:39 And so i have 2x minus 3y plus 4z equals negative 3 because i didn't change the red one.

00:49 And then since i'm multiplying here, i get negative 2x plus 2y plus 4z equals 2.

00:59 If i combine those equations by adding, i'm going to get negative y plus 8 z equals negative 1.

01:11 I'm going to repeat that process.

01:13 This time i'm going to use that second equation, negative x plus y plus 2z, equals 1.

01:22 I'm going to pair it with the third equation 5x, minus 2y, minus 3z, equals 7.

01:32 My coefficients don't line up, so i'm going to multiply the top equation by 5 to give me negative 5x plus 5y plus 10 z equals 5.

01:46 The blue one did not change, so i'm just rewriting it underneath.

01:53 Now to combine those by adding, i'm going to get 3y plus 7 z equals 12.

02:12 Now what i've done is i have reduced this down to a system of two equations in two variables.

02:18 So i'm going to take the two equations i now have and put them together...

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