Question

Solve the system of equations below. State your answer in the form (x, y, z). In the area below show all steps, and make sure to check your answer.

{x - y + 2z = 6
{2z + y - 2z = -3
{-x - 2y + 3z = 7

Answer 1

The system of equations is solved by substituting and eliminating variables to find x = -7/5, y = -3, and z = 16/5. The solution is then checked against the original equations to ensure accuracy.

Step-by-step explanation:

To solve the system of equations given by:

• x - y + 2z = 6
• 2z + y - 2z = -3
• -x - 2y + 3z = 7

We observe that the second equation simplifies to y = -3 because the 2z and -2z cancel each other out.

With y known, we can substitute it into the first and third equations:

• x + 3 + 2z = 6 → x + 2z = 3
• -x - 6 + 3z = 7 → -x + 3z = 13

Now, we have two equations with two variables:

• x + 2z = 3
• -x + 3z = 13

Adding them eliminates x:

• 5z = 16 → z = 16/5

Substitute z back into one of the above two-variable equations to find x:

• x + 2(16/5) = 3 → x = 3 - 32/5 → x = -7/5

Now we have x = -7/5, y = -3, and z = 16/5. Checking these values in the original equations confirms that they satisfy all three. Therefore, the solution in the form (x, y, z) is (-7/5, -3, 16/5).