Question

Help in solving question 5 please. System needs to be solved using the elimination method. Thanks!

Answer 1

Given: A system of linear equations in three variables x, y, and z as follows-

`\begin{gathered} 2x+y-z=9 \\ -x+6y+2z=-17 \\ 5x+7y+z=4 \end{gathered}`

Required: To solve the system using elimination.

Explanation: Let the given system as-

`\begin{gathered} 2x+y-z=9\text{ ...}(1) \\ -x+6y+2z=-17\text{ ...}(2) \\ 5x+7y+z=4\text{ ...\lparen3\rparen} \end{gathered}`

We can solve the system by reducing the system to a system of 2 variables. Suppose we would like to remove the variable z.

Multiplying equation (1) by 2, adding to equation (2) as follows-

`\begin{gathered} (4x+2y-2z)+(-x+6y+2z)=18+(-17) \\ 3x+8y=1\text{ ...}(4) \end{gathered}`

Now, add equations (1) and (3) as follows-

`\begin{gathered} (2x+y-z)+(5x+7y+z)=9+4 \\ 7x+8y=13\text{ ...}(5) \end{gathered}`

Now, equations (4) and (5) represent a system of linear equations in two variables. Subtracting the equations as follows-

`\begin{gathered} (3x+8y)-(7x+8y)=1-13 \\ -4x=-12 \\ x=3 \end{gathered}`

Substituting x=3 in equation (4)-

`\begin{gathered} 9+8y=1 \\ 8y=-8 \\ y=-1 \end{gathered}`

Substituting x=3 and y=-1 in equation (1) as follows-

`\begin{gathered} 6-1-z=9 \\ z=-4 \end{gathered}`

Final Answer: The solution to the system is-

`\begin{gathered} x=3 \\ y=-1 \\ z=-4 \end{gathered}`