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.
Solve the system using elimination:
3x + 2y - z = 7
-3x + y + 2z = -14
3x + y - z = 10
3 answers
Can you add the first two equations?
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.
(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.