Chapter 11, Problem 18RE

To determine

Solve each system of equations using the method of substitution ot the method of elimination.

Answer to Problem 18RE

It is inconsistent.

Explanation of Solution

Given information:

Solve each system of equations using the method of substitution or the method of elimination. If the system has no solution, say that it is inconsistent.

  $\{\begin{array}{l}x-4y+3z=15\\ -3x+y-5z=-5\\ -7x-5y-9z=10\end{array}$

Calculation:

The given system of equations in three variables is given by the expressions given below.

  $\{\begin{array}{l}x-4y+3z=15\\ -3x+y-5z=-5\\ -7x-5y-9z=10\end{array}$

The given system consists of three variables from which one of the variables is required to be removed from the equations. This can be done by multiplying first equations by $15$ and the second equation by $-9$ and the third equation by $-5$ . The equations obtained are given below.

  $\{\begin{array}{l}15x-60y+45=225\\ 27x-9y+45z=45\\ 35x+25y+45z=-50\end{array}$

We can now obtain two equations in $x$ and $y$ which can be solved simultaneously by subtracting

Second equation from the first equation and by subtracting third equation from the first equation.

  $\{\begin{array}{l}27x-15x-9y-(-60y)+45z-45z-225\\ 35x-15x+25y-(-60y)+45z-45z=-50-225\end{array}$

  $\{\begin{array}{l}12x+51y=-180\\ 20x+85y=-275\end{array}$

The given system of equations can be solved by the method of elimination. The first equation can be multiplied by $5$ and second equation can be multiplied by $3$ in order make the coefficient of $x$ same in both the equations.

  $\{\begin{array}{l}12x+51y=-180)\times 5\\ (20x+85y=-275)\times 3\end{array}$

  $\{\begin{array}{l}60x+255y=-900\\ 60x+255y=-825\end{array}$

The coefficient of $x$ in above equation is same and therefore the solutions to the equations can be determined by method of elimination.

The equations can now be solved for y since the coefficient of x is same in both the equations.

  $60x+255y-60x-255y=-825-(-900)$

  $0=-825+900$

  $0=75$

The above obtained results say that $75$ is equal to zero which is incorrect. Hence, the given system of equation is inconsistent.