Resolved: "Write the system of linear equations in the..."

College
Algebra
Linear algebra
Forms of linear equations

emancipezN

Answered question

2021-02-25

Write the system of linear equations in the form

A x = b

and solve this matrix equation for x.

- x_{1} + x_{2} = 4

- 2 x_{1} + x_{2} = 0

Answer & Explanation

avortarF

Skilled2021-02-26Added 113 answers

Step 1 We start with this system of linear equations: $- x_{1} + x_{2} = 4$
$- 2 x_{1} + x_{2} = 0$ But we want this type of set up: $A X = b$ So, $[\begin{matrix} - 1 & 1 \\ - 2 & 1 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = [\begin{matrix} 4 \\ 0 \end{matrix}]$ Now, let's write the LINEAR COMBINATION FORM using the COLUMN VECTORS that are the coefficients of the matrix A: $x_{1} [\begin{matrix} - 1 \\ - 2 \end{matrix}] + x_{2} [\begin{matrix} 1 \\ 1 \end{matrix}] = [\begin{matrix} 4 \\ 0 \end{matrix}]$

Now, let's find the augmented matrix to change it to ROW ECHELON FORM using GAUSSIAN ELIMINATION. DO NOT USE GAUSS-JORDAN ELIMINATION

Step 2 $A = [\begin{matrix} - 1 & 1 & - 4 \\ - 2 & 1 & 0 \end{matrix}]$

Step 3 $A = [\begin{matrix} - 1 & 1 & 4 \\ 0 & - 1 & - 8 \end{matrix}]$

Step 4 $A = [\begin{matrix} 1 & - 1 & - 4 \\ 0 & - 1 & - 8 \end{matrix}]$

Step 5 $A = [\begin{matrix} 1 & - 1 & - 4 \\ 0 & - 1 & - 8 \end{matrix}]$

Then, we get: $x_{1} - x_{2} = - 4$
$x_{2} = 8$

Step 6

$x_{1} - x_{2} = - 4$
$x_{2} = 8,$
$x_{1} - 8 = - 4$
$x_{2} = 8,$ So then we get: $x_{1} = - 4 + 8$
$x_{1} = 4$

Step 7

This is your solution: $x_{1} = 4$
$x_{2} = 8$

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