Question

Solve the system using elimination:

3x + 2y - z = 7
-3x + y + 2z = -14
3x + y - z = 10

Answer 1

Can you add the first two equations?

Answer 2

To solve this system using elimination, we will add the first equation to the third equation in order to eliminate the "x" term.

(3x + 2y - z) + (3x + y - z) = 7 + 10

This simplifies to:
6x + 3y - 2z = 17

Next, we will add this new equation to the second equation in order to eliminate the "x" term.

(6x + 3y - 2z) + (-3x + y + 2z) = 17 - 14

This simplifies to:
4y = 3

Dividing both sides of this equation by 4, we get:
y = 3/4

Now that we have the value of "y", we can substitute it back into one of the original equations to solve for "x". Let's use the second equation.

-3x + (3/4) + 2z = -14

Multiplying through by 4 to clear the fraction, we get:
-12x + 3 + 8z = -56

Rearranging this equation, we get:
-12x + 8z = -59

Next, we will substitute the value of "y" into the third equation to solve for "z".

3x + (3/4) - z = 10

Multiplying through by 4 to clear the fraction, we get:
12x + 3 - 4z = 40

Rearranging this equation, we get:
12x - 4z = 37

Now, we have a system of two equations:

-12x + 8z = -59
12x - 4z = 37

We can solve this system by adding the two equations together to eliminate the "x" term.

(-12x + 8z) + (12x - 4z) = -59 + 37

This simplifies to:
4z = -22

Dividing both sides of this equation by 4, we get:
z = -22/4
z = -11/2

Now that we have the value of "z", we can substitute it back into one of the original equations to solve for "x". Let's use the first equation.

3x + 2(3/4) - (-11/2) = 7

Simplifying this equation, we get:
3x + 3/2 + 11/2 = 7
3x + 14/2 = 7
3x + 7 = 7
3x = 0
x = 0

Therefore, the solution to the system of equations is x = 0, y = 3/4, and z = -11/2.

Answer 3

Yes, we can add the first two equations together to eliminate the "x" term.

(3x + 2y - z) + (-3x + y + 2z) = 7 + (-14)

This simplifies to:
3y + z = -7

Now, we have the new equation of 3y + z = -7 and the third equation of 3x + y - z = 10.

To solve for "y" and "z" in the new equation 3y + z = -7, we can isolate "z" by subtracting 3y from both sides:

z = -7 - 3y

Substituting this value of "z" into the equation 3x + y - z = 10:

3x + y - (-7 - 3y) = 10
3x + y + 7 + 3y = 10
3x + 4y + 7 = 10

Moving the constants to one side, we get:
3x + 4y = 3

We now have a system of two equations:

3x + 4y = 3
3x + y - z = 10

We can solve this system by adding the two equations together to eliminate the "x" term:

(3x + 4y) + (3x + y - z) = 3 + 10

This simplifies to:
6x + 5y - z = 13

So the solution to the system of equations is 6x + 5y - z = 13, 3y + z = -7, and 3x + y - z = 10.