To solve the system using elimination, we need to add or subtract the equations in a way that eliminates one variable.
We can eliminate the x term by multiplying the first equation by -1 and adding it to the second equation:
(-1)(3x + 2y - z = 7)
-3x + y + 2z = -14
This gives us:
-3x - 2y + z = -7
-3x + y + 2z = -14
Next, we can add the second equation to the third equation:
-3x + y + 2z = -14
+3x + y - z = 10
This gives us:
2y + z = -4
Subtracting equation 2 from equation 3 allows us to eliminate x.
Simplifying equation 3 gives us:
2y + z = -4
Now, let's solve these two equations simultaneously.
From equation 4:
z = -4 - 2y
Substitute this value into equation 1:
-3x - 2y + z = -7
-3x - 2y + (-4 - 2y) = -7
Simplifying this equation gives us:
-3x - 4y = -3
To further simplify, let's multiply equation 5 by -3 and add it to equation 6:
(-3)(-3x - 4y = -3)
9x + 12y = 9
-3x - 4y = -3
This gives us:
6x + 8y = 6
-3x - 4y = -3
Now let's multiply equation 7 by 2 and add it to equation 8 to eliminate y:
2(6x + 8y = 6)
-3x - 4y = -3
This gives us:
12x + 16y = 12
-3x - 4y = -3
Adding these two equations gives us:
9x + 12y - 3x - 4y = 9 - 3
6x + 8y = 6
-3x - 4y = -3
This simplifies to:
6x + 8y = 6
-3x - 4y = -3
Multiplying equation 9 by 2 gives us:
12x + 16y = 12
-6x - 8y = -6
Adding equation 10 to equation 11 gives us:
12x + 16y - 6x - 8y = 12 - 6
6x + 8y = 6
6x + 8y = 6
We have reached the same equation twice, which means this system of equations has infinite solutions.
Solve the system using elimination:
3x + 2y - z = 7
-3x + y + 2z = -14
3x + y - z = 10
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