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# 1: Systems of Linear Equations- Algebra

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## Chapter1Systems of Linear Equations: Algebra

##### Note

Solve a system of linear equations algebraically in parametric form.

This chapter is devoted to the algebraic study of systems of linear equations and their solutions. We will learn a systematic way of solving equations of the form

$\left\{\begin{array}{rrrrrrrrrrrrr} 3x_1 &+& 4x_2 &+& 10x_3 &+& 19x_4 &-& 2x_5 &-& 3x_6 &=& 141\\ 7x_1 &+& 2x_2 &-& 13x_3 &-& 7x_4 &+& 21x_5 &+& 8x_6 &=& 2567\\ -x_1 &+& 9x_2 &+& \frac 32x_3 &+& x_4 &+& 14x_5 &+& 27x_6 &=& 26\\ \frac 12x_1 &+& 4x_2 &+& 10x_3 &+& 11x_4 &+& 2x_5 &+& x_6 &=& -15 \end{array}\right. \nonumber$

In Section 1.1, we will introduce systems of linear equations, the class of equations whose study forms the subject of linear algebra. In Section 1.2, will present a procedure, called row reduction, for finding all solutions of a system of linear equations. In Section 1.3, you will see hnow to express all solutions of a system of linear equations in a unique way using the parametric form of the general solution.

1: Systems of Linear Equations- Algebra is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Dan Margalit & Joseph Rabinoff via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.