Solve the system of equations: x - y - z = 2; x + 2y - 2z = 3; 3x - 2y - 4z = 5?
well, you have:
1- x-y-z=2
2- x+2y-2z=3
3- 3x-2y-4z=5
multiply both sides of the equation 1 by 2:
x-y-z=2 ===> 2x-2y-2z=4
add this by equation 2:
2x-2y-2z=4
x+2y-2z=3
==> 3x - 4z = 7 (*)
now, multiple both sides of equation 1 by -2:
x-y-z=2
==> -2x + 2y + 2z = -4
add this by equation 3:
-2x + 2y + 2z = -4
3x-2y-4z=5
===> x - 2z = 1
great. now you have
x - 2z = 1
and
3x - 4z = 7
so , multiple both sides of the first one by -2:
-2x + 4z = -2
3x - 4z = 7
===> (add them together) x = 5
3x - 4z = 7 ===(x=5)=> 15 - 4z = 7 ==> 4z = 8 ==> z=2
x-y-z= 2 ===> 5-y-2 = 2 ==> y=1
so, the answer is x=5, y=1, z=2
The solution to the system of equations is x = 5, y = 1, and z = 2.
To solve the system of equations, we will use the method of elimination or substitution. Let's start with the elimination method:
Step 1: Multiply equation 1 by 2:
2(x - y - z) = 2(2)
This gives us: 2x - 2y - 2z = 4
Step 2: Add equation 2 to the result of step 1:
(2x - 2y - 2z) + (x + 2y - 2z) = 4 + 3
This simplifies to: 3x - 4z = 7 (*)
Step 3: Multiply equation 1 by -2:
-2(x - y - z) = -2(2)
This gives us: -2x + 2y + 2z = -4
Step 4: Add equation 3 to the result of step 3:
(-2x + 2y + 2z) + (3x - 2y - 4z) = -4 + 5
This simplifies to: x - 2z = 1
Now we have two equations:
3x - 4z = 7
x - 2z = 1
To eliminate x, we can multiply the second equation by 3:
3(x - 2z) = 3(1)
This gives us: 3x - 6z = 3
Now we can subtract this equation from the first equation:
(3x - 6z) - (3x - 4z) = 3 - 7
Simplifying further, we get: 2z = -4
Dividing both sides by 2 gives us: z = -2
Substitute the value of z into the second equation:
x - 2(-2) = 1
Simplifying, we get: x + 4 = 1
Subtracting 4 from both sides gives us: x = -3
Substitute the values of x and z into the first equation to solve for y:
-3 - y - (-2) = 2
Simplifying, we get: -3 - y + 2 = 2
Combining like terms, we get: -y = 3
Dividing both sides by -1 gives us: y = -3
So the solutions to the system of equations are:
x = -3
y = -3
z = -2