Question

Test for consistency, the given system of equations and solve, if it is consistent.
7x+2y+3z=16
2x+11y+5z=25
x+3y+4z=13

Answer 1

Answer:

The equations are consistent.

Step-by-step explanation:

We are given following equations for which we have to check the consistency:

7x + 2y + 3z = 16

2x + 11y + 5z = 25

x + 3y + 4z = 13

We know that set of equations are said be consistent if they have at least one or many solutions.

Now writing the augmented matrix for above equations

$\left[\begin{array}{cccc}7&2&3&:16\\2&11&5&:25\\1&3&4&:13\end{array}\right]$

Now change place of R₃ and R₁

$\left[\begin{array} {cccc}1&3&4&:13\\2&11&5&:25\\7&2&3&:16\end{array}\right]$

Now perform the row operations R₂ - 2R₂ and R₃ - 7R₁ on the above augmented matrix. After performing these operations we will get the following matrix

$\left[\begin{array} {cccc}1&3&4&:13\\0&5&-3&:-1\\0&-19&-25&:-75\end{array}\right]$

Now perform the row operation R₂/5

$\left[\begin{array} {cccc}1&3&4&:13\\0&1&-\frac{3}{5}&:-[\frac{1}{5}\\0&-19&-25&:-75\end{array}\right]$

Now perform the row operations R₃ +19R₂  on the above matrix. After performing these operations we will get the following matrix

$\left[\begin{array} {cccc}1&3&4&:13\\0&1&-\frac{3}{5}&:-\frac{1}{5}\\0&0&-25 - \frac{57}{3}&:-75 + \frac{19}{5}\end{array}\right]$

Now, we can write equation for last row as

$(-25 - \frac{57}{3})z = -75 + \frac{19}{5}$

$(-\frac{25*3+57}{3})z = -\frac{75*5 + 19}{5}$

$(-\frac{132}{3})z = -\frac{394}{5}$

$(44z = \frac{394}{5}$

Which can solve be solved as

$z = \frac{197}{110}$

Now we can substitute $z = \frac{197}{110}$ into the above row and find y and similarly we can find x.

This process generates a unique solution for the system of equations, therefore our system of equations is consistent.