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{$\begin{gathered}\sf \begin{cases}-15y-10z=-30 \\ z=-(-40+13y)/(4) \end{cases} \end{gathered}$}`

Substitute for one of the equations.

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

Reduce the expression to the least term.

`\boldsymbol{\sf{-15y+5*(-40+13y)/(2)=-30 }}`

Multiply both sides by the common denominator.

`\boldsymbol{\sf{-15y*2+5*((-40+13y)*2)/(2)=-30*2 }}`

Simplify the fractions.

`\boldsymbol{\sf{-15y*2+5*(-40+13y)=-30*2 }}`

Aplicar la ley multiplicativa de distribución.

`\boldsymbol{\sf{-15y*2-200+65y=-30*2 }}`

Multiply the monomial.

`\boldsymbol{\sf{-30y-200+65y=-30*2 }}`

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=(140)/(35) \ \ \to \ Split }}`

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

Substitute for one of the equations.

`\boldsymbol{z=-(-40+13*4)/(4) \ \to \ Multiply }`

`\boldsymbol{z=-(-40+52)/(4) \ \to \ Add }`

`\boldsymbol{z=-(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*4-(-3)}`

Calculate

`\bf{x=0}`

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