Question

Solve
6x+3y-4z=24
6x+5y+2z=14
x+3y+z=9
​

Answer 1

The first one is x=4.16z
The second one is x=1.16z
The third one is x=6z

Answer 2

   $\large\displaystyle\text{ \begin{gathered}\sf \bf{\begin{cases}6x+3y-4z=24 \\6x+5y+2z=14 \\ x+3y+z=9 \end{cases} \ \ \longmapsto \ \ \ Your \ exercise} \end{gathered} }$

Reorder

                    $\large\displaystyle\text{ \begin{gathered}\sf \begin{cases}6x+3y-4z=24 \\6x+5y+2z=14 \\ x=9-3y-z \end{cases} \end{gathered} }$

Substitute one of the equations:

                          $\large\displaystyle\text{ \begin{gathered}\sf \begin{cases}6(9-3y-z)+3y-4z=24 \\ 6(9-3y-z)+5y+2z=14 \end{cases} \end{gathered} }$

Apply the multiplicative law of distribution.

                    $\large\displaystyle\text{ \begin{gathered}\sf \begin{cases}54-18y-6z +3y-4z=24 \\ 54-18y-6z+5y+2z=14 \end{cases} \end{gathered} }$

Combine as terms.

                                $\large\displaystyle\text{ \begin{gathered}\sf \begin{cases}54-15y-10z=24 \\ 54-13y-4z=14 \end{cases} \end{gathered} }$

Rearrange the unknown terms on the left side of the equation.

                   $\large\displaystyle\text{ \begin{gathered}\sf \begin{cases}-15y-10z=24-54 \\ 54-13y-4z=14-54 \end{cases} \ \longmapsto \ Subtraction \end{gathered} }$

                              $\large\displaystyle\text{ \begin{gathered}\sf \begin{cases}-15y-10z=-30 \\ -13y-4z=-40 \end{cases} \end{gathered} }$

Rearrange like terms on the same side of the equation.

                                  $\large\displaystyle\text{ \begin{gathered}\sf \begin{cases}-15y-10z=-30 \\ -4z=-40+ 13y\end{cases} \end{gathered} }$

Divide both sides of the equation by the coefficient of the variable.

                              $\large\displaystyle\text{ \begin{gathered}\sf \begin{cases}-15y-10z=-30 \\ z=-\frac{-40+13y}{4} \end{cases} \end{gathered} }$

Substitute for one of the equations.

         $\boldsymbol{\sf{-15y-10(-\dfrac{-40+13y}{4})=-30 }}$

Reduce the expression to the least term.

                       $\boldsymbol{\sf{-15y+5\times\dfrac{-40+13y}{2}=-30 }}$

Multiply both sides by the common denominator.

                   $\boldsymbol{\sf{-15y\times2+5\times\dfrac{(-40+13y)\times2}{2}=-30\times2 }}$

Simplify the fractions.

                      $\boldsymbol{\sf{-15y\times2+5\times(-40+13y)=-30\times2 }}$

Aplicar la ley multiplicativa de distribución.

                 $\boldsymbol{\sf{-15y\times2-200+65y=-30\times2 }}$

Multiply the monomial.

                       $\boldsymbol{\sf{-30y-200+65y=-30\times2 }}$

Calculate the product or coefficient.

                                $\boldsymbol{\sf{-30y-200+65y=-60 }}$

Combine as terms.

                             $\boldsymbol{\sf{35y-200=-60 }}$

Rearrange the unknown terms on the left side of the equation.

                                    $\boldsymbol{\sf{35y=-60+200 }}$

Calculate the sum or difference.

                              $\boldsymbol{\sf{35y=140 }}$

Divide both sides of the equation by the coefficient of the variable.

                                 $\boldsymbol{\sf{y=\dfrac{140}{35} \ \ \to \ Split }}$

                                 $\boldsymbol{\sf{y=4}}$

Substitute for one of the equations.

                          $\boldsymbol{z=-\dfrac{-40+13\times4}{4} \ \to \ Multiply }$

                              $\boldsymbol{z=-\dfrac{-40+52}{4} \ \to \ Add }$

                              $\boldsymbol{z=-\dfrac{12}{4} \ \to \ Split }$

                              $\boldsymbol{z=-3 }$

The system solution is:

                                  $\boldsymbol{\sf{\begin{cases} y=4 \\ z=-4 \end{cases} }}$

Substitute for one of the equations.

                                 $\bf{x=9-3\times4-(-3)}$

Calculate

                                        $\bf{x=0}$

              $\huge \red{\boxed{\boldsymbol{\sf{\green{Answer \ \longmapsto \ \begin{cases} x=0 \\ y=4 \\ z=-3 \end{cases}}}}}}$