Question

3x + 2y - z= 7This system has three equations: {-3x + y + 2z = -143x + y -z = 101. Add the first two equations to get a new equation.2. Add the second two equations to get a new equation.3. Solve the system of your two new equations.4. What is the solution to the original system of equations?

Answer 1

We are given the following system of equations:

`\begin{gathered} 3x+2y-z=7,(1) \\ -3x+y+2z=-14,(2) \\ 3x+y-z=10,(3) \end{gathered}`

1. We will add equation (1) and equation (2), we get:

`3x+2y-z-3x+y+2z=7-14`

Now we add like terms:

`3y+z=-7,(4)`

We get a new equation which we called equation (4).

2. Now, we will add equation (2) and equation (3), we get:

`-3x+y+2z+3x+y-z=-14+10`

Now, we add like terms:

`2y+z=-4,(5)`

We get a new equation that we called equation (5).

3. equation (4) and equation (5) form a new system of equations:

`\begin{gathered} 3y+z=-7,(4) \\ 2y+z=-4,(5) \end{gathered}`

To solve the system we will subtract equation (5) from equation (4), this way we will be able to cancel out the variable "z". Like this:

`3y+z-2y-z=-7+4`

Now, we add like terms:

`y=-3`

Therefore, the value of "y" is -3.

Now, we substitute the value of "y" in equation (4), we get:

`3(-3)+z=-7`

Solving the product:

`-9+z=-7`

Now, we add 9 to both sides:

`\begin{gathered} -9+9+z=-7+9 \\ z=2 \end{gathered}`

Therefore, we have the following values from the new system:

`\begin{gathered} y=-3 \\ z=2 \end{gathered}`

4. To determine the values of the original system we will substitute the values of "y" and "z" in equation (1):

`3x+2(-3)-(2)=7`

Solving the operations we get:

`\begin{gathered} 3x-6-2=7 \\ 3x-8=7 \end{gathered}`

Now, we will add 8 to both sides:

`\begin{gathered} 3x-8+8=7+8 \\ 3x=15 \end{gathered}`

Now, we will divide both sides by 3:

`x=(15)/(3)=5`

Therefore, the solution of the system is:

`(x,y,z)=(5,-3,2)`