# Use Gaussian elimination to find the complete solution to the system of given equations, or show that none exists. {(5x+8y-6z=14),(3x+4y-2z=8),(x+2y-2z=3):}

Question
Equations
Use Gaussian elimination to find the complete solution to the system of given equations, or show that none exists.
$$\displaystyle{\left\lbrace\begin{array}{c} {5}{x}+{8}{y}-{6}{z}={14}\\{3}{x}+{4}{y}-{2}{z}={8}\\{x}+{2}{y}-{2}{z}={3}\end{array}\right.}$$

2020-11-08
Step 1
Consider the following system of linear equations:
5x+8y-6z=14
3x+4y-2z=8
x+2y-2z=3
Convert above system of linear equations into augmented matrix form:
$$\displaystyle{\left[\begin{array}{ccc|c} {5}&{8}&-{6}&{14}\\{3}&{4}&-{2}&{8}\\{1}&{2}&-{2}&{3}\end{array}\right]}$$
Step 2
Transform the above matrix into reduced row echelon form:
$$\displaystyle{R}_{{2}}\rightarrow\frac{{5}}{{3}}{R}_{{2}}-{R}_{{1}}$$
$$\displaystyle{R}_{{3}}\rightarrow{5}{R}_{{3}}-{R}_{{1}}$$
$$\displaystyle{\left[\begin{array}{ccc|c} {5}&{8}&-{6}&{14}\\{0}&-\frac{{4}}{{3}}&\frac{{8}}{{3}}&-\frac{{2}}{{3}}\\{0}&{2}&-{4}&{1}\end{array}\right]}$$
$$\displaystyle{R}_{{3}}\rightarrow\frac{{2}}{{3}}{R}_{{3}}+{R}_{{2}}$$
$$\displaystyle{\left[\begin{array}{ccc|c} {5}&{8}&-{6}&{14}\\{0}&-\frac{{4}}{{3}}&\frac{{8}}{{3}}&-\frac{{2}}{{3}}\\{0}&{0}&{0}&{0}\end{array}\right]}$$
Step 3
Hence, the solution of a system of linear equations does not exist.

### Relevant Questions

Use Gaussian elimination to find the complete solution to the system of given equations, or show that none exists
$${(5x,+,8y,-,6y,=,14),(3x,+,4y,-,2z,=,8),(x,+,2y,-,2z,=,3):}$$
Use Gaussian elimination to find the complete solution to the system of given equations, or show that none exists.
$$\displaystyle{\left\lbrace\begin{array}{c} {2}{x}-{4}{y}+{z}={3}\\{x}-{3}{y}+{z}={5}\\{3}{x}-{7}{y}+{2}{z}={12}\end{array}\right.}$$
Solve the given system of equations by matrix equation.
5x-4y=4
3x-2y=3
Use elimination to solve the system of equation:
2x+2y=-2
3x-2y=12
Find values of a and b such that the system of linear equations has no solution.
x+2y=3
ax+by=-9
Solve the system. If the system does not have one unique solution, also state whether the system is onconsistent or whether the equations are dependent.
2x-y+z=-3
x-3y=2
x+2y+z=-7

Determine if (1,3) is a solution to the given system of linear equations.

$$5x+y=8$$

$$x+2y=5$$

Find the discriminant of each equation and determine whether the equation has (1) two nonreal complex solutions, (2) one real solution with a multiplicity of 2, or (3) two real solutions. Do not solve the equations. $$7x^{2} - 2x - 14 = 0$$
$$x+2y=8. x=-5$$