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Solving Linear Equations in Systems

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Questions and Answers

What is the correct value of $z$ from the given system of equations?

2 (correct)

Which equation corresponds to the second row after performing row operations?

$y + 3z = 5$ (correct)

$-3y + z = -5$

$-y + 3z = 5$

$0y + 2z = 6$

What equation must be solved to find the value of $y$?

$z - 2y = 1$

$2z = 4$

$x - 2y + 9 = 0$

$y + 6 = 5$ (correct)

What transformation was performed on the first row during Gaussian elimination?

<p>Addition of $R_1$ to $R_2$ (D)</p> Signup and view all the answers

What is the final value of $x$ after solving the equations?

<p>1 (A)</p> Signup and view all the answers

Flashcards

Gauss Elimination

A method to solve systems of linear equations by manipulating an augmented matrix representing those equations.

Augmented Matrix

A matrix combining the coefficients of variables and constants of a system of linear equations.

Row Operations

Elementary operations on the rows of a matrix, such as exchanging rows, multiplying by a non-zero scalar and adding a multiple of one row to another to simplify the matrix and solve the system.

Back Substitution

The process of solving for unknowns in a system of linear equations once the matrix has been simplified to an upper triangular form, beginning with the last equation and progressively determining the values of unknowns.

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System of Linear Equations

A set of two or more linear equations containing the same unknowns.

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Study Notes

Linear Equations Systems

Systems of linear equations can have one solution, no solution, or infinitely many solutions.
Methods to solve systems of linear equations include Gaussian elimination and substitution.

Solving Systems of Equations (Examples)

Example 1:
- x + y = 5
- y = 5 – x
- This system has infinitely many solutions, expressed parametrically as p = t; y = 5 – t.
Example 2:
- x + y = 2
- -x – y = 3
- This system has no solution.
Example 3:
- x + y = 2
- x – y = 4
- Solution: x = 3; y = -1
Example 4:
- x + y = 2
- 2x + 2y = 4
- This system has infinitely many solutions, expressed parametrically as x = t; y = 2 - t.

Gaussian Elimination

This method is used for systems of three or more variables.
It involves transforming the system of equations into a triangular form using row operations to simplify the matrix of coefficients.
Example:
- x – 2y + 3z = 9
- -x + 3y = -4
- 2x – 5y + z = 17
- Solving the above equations by Gaussian Elimination yields x = 1, y = 2, z = 2

Homework Assignment

Solve the following system of linear equations: x + 2y + 3z = 3, 2x + 5y + 7z = 6, 3x + 7y + 8z = 5.

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Description

This quiz covers the solution of systems of linear equations, including methods such as Gaussian elimination and substitution. You will explore examples that illustrate systems with one solution, no solution, or infinitely many solutions. Test your understanding of these concepts through practical examples.