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Solve the system of equations using Gaussian elimination 4x-8y=12 -x+2y=-3

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asked 2021-02-04
Solve the system of equations using Gaussian elimination
4x-8y=12
-x+2y=-3

Answers (1)

2021-02-05
Step 1
The system of equations is given by,
4x-8y=12
-x+2y=-3
Step 2
Find the solution of system of equations, it is need to write an equivalent matrix equation AX=B .
$$\displaystyle{\left[\begin{array}{cc} {4}&-{8}\\-{1}&{2}\end{array}\right]}\cdot{\left[\begin{array}{c} {x}\\{y}\end{array}\right]}={\left[\begin{array}{c} {12}\\-{3}\end{array}\right]}$$
Find the inverse matrix by using Gauss-Jordan elimination method.
That is, apply the row equivalent operations to the matrix.
Consider the matrix $$\displaystyle{A}={\left[\begin{array}{cc} {4}&-{8}\\-{1}&{2}\end{array}\right]}$$.
Form of an augment matrix, in order to find the inverse matrix.
That is, form an matrix contains of A on the left side and the $$\displaystyle{2}\times{2}$$ identify matrix on the right side.
$$\displaystyle{\left[\begin{array}{cc|c} {4}&-{8}&{12}\\-{1}&{2}&-{3}\end{array}\right]}$$
Step 3
Divide the first row by 4 as follows.
$$\displaystyle{\left[\begin{array}{cc|c} {1}&-{2}&{3}\\-{1}&{2}&-{3}\end{array}\right]}$$ New row $$\displaystyle{1}=\frac{{1}}{{4}}$$ row 1
Add the first and second rows as follows.
$$\displaystyle{\left[\begin{array}{cc|c} {1}&-{2}&{3}\\{0}&{0}&{0}\end{array}\right]}$$ New row 2 = row 1 + row 2
Thus, the system of equations has a solution set is $$\displaystyle{\left\lbrace{x}-{2}{y}={3}.\right.}$$.
Step 4
Answer:
The system of equations has a solution set $$\displaystyle{\left\lbrace{x}-{2}{y}={3}.\right.}$$.

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