Solve each system. 3 a+5 b-3 c =-4 a-3 b+c =6 -4 a+6 b+2 c =-6

Name: Problem 11, Chapter 5: Beginning and Intermediate Algebra
Uploaded: 2021-03-17T18:42:25-08:00
Duration: 6 min 54 s
Channel: Debasish Das
Description: Solve each system. 3 a+5 b-3 c =-4 a-3 b+c =6 -4 a+6 b+2 c =-6

step 1 Question number 11 here are to solve the equation the equation is given as a a plus 5b minus 3c

Solve each system. 3 a+5 b-3 c =-4 a-3 b+c =6 -4...

Key Concepts

Matrix Methods (Gaussian Elimination)

Matrix methods, such as Gaussian elimination, represent the system of equations in an augmented matrix form. Through a series of row operations, this matrix is transformed into a row-echelon form (or reduced row-echelon form), from which the solutions can be read directly. This method is systematic and particularly effective for solving larger systems.

Substitution Method

The substitution method involves solving one of the equations for one variable and then substituting this expression into the other equations. This method gradually reduces the number of unknowns and simplifies the system of equations, making it easier to solve. It is particularly useful when one equation in the system is already solved or can be easily solved for a variable.

Consistency and Unique Solutions

When solving a system of linear equations, it is important to assess the system’s consistency. A system can have a unique solution, infinitely many solutions, or no solution at all. Determining the type of solution involves checking whether the procedures (such as elimination or matrix reduction) lead to a valid, unique answer, a contradictory equation, or a dependency among the equations.

System of Linear Equations

A system of linear equations is a collection of equations where each equation is linear in the unknown variables. The aim is to find a set of values that simultaneously satisfies all the equations. These systems are foundational in algebra and have applications in various fields including engineering, physics, and economics.

Elimination Method

The elimination method is a technique used to solve systems of linear equations by removing one variable at a time. This is accomplished by adding or subtracting equations (or multiples of equations) so as to cancel out one of the variables, thus reducing the system to a simpler one. Once a variable is eliminated, the process is repeated until the values of all variables can be determined.

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