Question

Systems with3 VARIABLESx-5y+3z=393x+8y-z=-192x-6y-5z=-29

Answer 1

We need to solve one system with three variables and three equations. To do that the first step is to isolate one variable on one of the equations and substitute that expression on the others. Let's isolate the "x" variable on the first equation.

`\begin{gathered} x-5y+3z=39 \\ x=5y-3z+39 \end{gathered}`

We can use that value of "x" on the other equations and then we will have only two equations and two variables.

`\begin{gathered} 3\cdot(5y-3z+39)+8y-z=-19 \\ 15y-9z+117+8y-z=-19 \\ 23y-10z=-19-117 \\ 23y-10z=-136 \end{gathered}`
`\begin{gathered} 2\cdot(5y-3z+39)-6y-5z=-29 \\ 10y-6z+78-6y-5z=-29 \\ 4y-11z=-29-78 \\ 4y-11z=-107 \end{gathered}`

We can create a new system of equations with these two expressions.

`\mleft\{\begin{aligned}23y-10z=-136 \\ 4y-11z=-107\end{aligned}\mright.`

If we multiply the first equation by "-4" and the second by "23" we can eliminate the "y" variable.

`\begin{gathered} \{\begin{aligned}(23y-10z=-136)\cdot(-4) \\ (4y-11z=-107)\cdot23\end{aligned} \\ \{\begin{aligned}-92y+40z=544 \\ 92y-253z=-2461\end{aligned} \\ -92y+92y+40z-253z=544-2461 \\ -213z=-1917 \\ z=(-1917)/(-213)=9 \end{gathered}`

If we use the value of "z" on either of the equations we should be able to find he value of y.

`\begin{gathered} 4y-11\cdot(9)=-107 \\ 4y-99=-107 \\ 4y=-107+99 \\ 4y=-8 \\ y=(-8)/(4)=-2 \end{gathered}`

Using these values on the expression for "x", we have:

`\begin{gathered} x=5y-3z+39 \\ x=5\cdot(-2)-3\cdot9+39 \\ x=-10-27+39 \\ x=2 \end{gathered}`

The solution to this system is x=2, y=-2 and z = 9