System of Equation

[Tiger-T]

Need to Solve the system of equations. Let z be the parameter.
-3x + 5y + 4z = -5
-4x + y -4z = -9
The solutions include
A) 40/17 - 24/17z, 7/17- 28/17z,z
B) 40/17 - 24/17z, -9 - 4z,z
C) 40/17 - 24/17z - 9 + 4z,z
D) 40 - 24z, 7 + 40z,z

Can some guide me on to solve such a question?

 

[soroban]

Hello, Tiger-T!

Solve the system of equations. . Let \(\displaystyle z\) be the parameter.

. . \(\displaystyle \begin{array}{ccccc}-3x + 5y + 4z &=& -5 & [1] \\ -4x + y -4z &=& -9 & [2] \end{array}\)

The solutions include

.\(\displaystyle (A)\;\begin{Bmatrix}x &=& \frac{40}{17} - \frac{24}{17}z \\ \\[-3mm] y &=& \frac{7}{17}- \frac{28}{17}z \\ z &=& z \end{Bmatrix} \quad (B)\;\begin{Bmatrix}x &=& \frac{40}{17} - \frac{24}{17}z \\ \\[-4mm] y &=& \text{-}9 - 4z \\ z &=&z \end{Bmatrix} \quad (C)\;\begin{Bmatrix}x &=& \frac{40}{17} - \frac{24}{17}z \\ \\[-4mm] y &=& \text{-} 9 + 4z \\ x &=& z \end{Bmatrix} \quad (D)\;\begin{Bmatrix}x &=& 40 - 24z \\ y &=& 7 + 40z \\ z &=&z \end{Bmatrix}\)

\(\displaystyle \begin{array}{ccccccc}\text{Multiply [2] by -5:} & 20x - 5y + 20z &=& 45 \\ \text{Add [1]:} & -3x + 5y + 4z &=& -5 \end{array}\)

\(\displaystyle \text{We have: }\:17x + 24z \:=\:40 \quad\Rightarrow\quad x \:=\:\frac{40-24z}{17} \quad\Rightarrow\quad x\;=\;\frac{40}{17} - \frac{24}{17}z\)

\(\displaystyle \text{Substitute into [2]: }\:-4\left(\frac{40-24z}{17}\right) + y - 4z \:=\:-9\)

. . \(\displaystyle \text{Solve for }y:\;y \:=\:\frac{7-28z}{17} \quad\Rightarrow\quad y \:=\:\frac{7}{17} - \frac{28}{17}z\)


\(\displaystyle \text{Therefore: }\;\begin{Bmatrix}x &=& \frac{40}{17} - \frac{24}{17}z \\ \\[-3mm]y &=& \frac{7}{17} - \frac{28}{17}z \\ z &=& z \end{Bmatrix} \;\hdots\; \text{ answer (A)}\)

 

[mmm4444bot]

"Spoon feeding, in the long run, teaches us nothing but the shape of the spoon."

~ Edward Morgan Forster