Extend the concepts of this section to solve each system. Write the solution in the form (a, b,

step 1 Okay, moving on to question number 61. Four equations, four unknowns. So, equation number one is

Extend the concepts of this section to solve each system....

Key Concepts

Systems of Linear Equations

This concept involves solving multiple equations simultaneously where each equation is linear in the unknowns. The goal is to find values of the variables that satisfy every equation at the same time, ensuring the system is consistent or identifying cases when it is over- or under-determined.

Gaussian Elimination

Gaussian elimination is a systematic method used to solve systems of linear equations. It involves using elementary row operations on the augmented matrix of the system to transform it into an upper triangular form, from which the solutions can be found by back-substitution.

Matrix Representation

Matrix representation simplifies a system of linear equations by organizing its coefficients and constants into a matrix format. This method provides a clear structure for applying row reduction techniques and leveraging computational algorithms to systematically solve the system.

Row Operations

Row operations are the fundamental manipulations—such as row switching, scaling, and adding multiples of rows—that are applied during Gaussian elimination. These operations help transform the matrix into a simpler form without changing the solution set of the system.

Transcript

00:04 Okay, moving on to question number 61.

00:20 Four equations, four unknowns.

00:23 So, equation number one is 3a plus 4b plus c minus d is equal to negative 7.

00:29 And equation number two is negative 3a minus 4b minus c plus d equals 1.

00:38 So i'm going to add equation number 1 and 2.

00:41 So let me write equation number 1 and 2 together.

00:44 Okay so question number one is 3a plus 4 b plus c minus d is equal to negative 7 this is equation number 1 and negative 3a minus 2 b minus c plus d is equal to 1 d is equal to 1 this is equation number two so i'm going to add question number one and do so question number one plus equation number two as we notice here three a minus three a cancel out c minus c cancel out negative d positive d cancel out so we end up getting two b is equal to negative which implies b equal to negative three pretty simple b is equal to negative three so very easily we were able to find the value of b okay and now i'm going to add question number uh equation number two and four okay so i'm going to write equation number two here so question number two is negative three a minus 2b minus c plus d is equal to 1 and equation number 4 is 2a plus b plus c minus d is equal to 2 so this is my equation number 2 which i already wrote on top this is my equation number 4 from the question so i'm going to add 2 and 4 so equation number 2 and equation number 4 negative 3a plus 2a is going to be negative 1a okay then negative 2b plus 2 plus b is negative 1b and negative c positive c cancel out positive d negative d cancel out and you get 1 plus 2 plus 2 is equal to 3.

03:38 And we already know that b is negative 3.

03:41 So i'm going to substitute b in this.

03:43 Okay.

03:44 So substitute b in this equation.

03:48 You get, i'm going to draw an arrow here.

03:55 You put that value there.

03:59 I get negative a minus b is negative 3 is equal to 3.

04:11 Okay.

04:12 So which means negative a, negative times negative is positive.

04:18 So negative a plus three is equal to zero, which means, not zero, three, which means a is equal to zero.

04:31 A is equal to zero.

04:35 So we found the value of two variables, a is equal to zero.