4x+2y-3z=7 2x+4y+z=35 6x+2y-2z=20 solve the system

#1 + 3(#2) ----> 10x + 14y = 112 or 5x + 7y = 56 (#4)

#3 + 2(#2) ---> 10x + 10y = 90 or 5x + 5y = 45 (#5)

#4 - #5 ---> 2y = 11
y = 5.5
in #5 --> x + 5.5 = 9
x = 3.5

in #2
2(3.5) + 4(5.5) + z = 35
z = 6

I will leave it up to you to check those answers.

To solve the system of equations:

4x + 2y - 3z = 7 ...(Equation 1)
2x + 4y + z = 35 ...(Equation 2)
6x + 2y - 2z = 20 ...(Equation 3)

We can use the method of elimination or substitution. Let's solve the system using elimination.

1. Multiply Equation 1 by 2: 8x + 4y - 6z = 14

Now, we can eliminate the y term by subtracting Equation 2 from the above equation.

(8x + 4y - 6z) - (2x + 4y + z) = 14 - 35
6x - 7z = -21 ...(Equation 4)

2. Multiply Equation 2 by 2: 4x + 8y + 2z = 70

Again, we can eliminate the y term by subtracting Equation 3 from the above equation.

(4x + 8y + 2z) - (6x + 2y - 2z) = 70 - 20
-2x + 10y + 4z = 50 ...(Equation 5)

3. Now, we have two equations:

6x - 7z = -21 ...(Equation 4)
-2x + 10y + 4z = 50 ...(Equation 5)

4. To eliminate the x term, multiply Equation 4 by -1 and Equation 5 by 3:

-6x + 7z = 21 ...(Equation 6)
-6x + 30y + 12z = 150 ...(Equation 7)

5. Add Equation 6 and Equation 7 to eliminate the x term:

(-6x + 7z) + (-6x + 30y + 12z) = 21 + 150
37z + 30y = 171 ...(Equation 8)

6. Now, we have two equations:

37z + 30y = 171 ...(Equation 8)
-2x + 10y + 4z = 50 ...(Equation 5)

7. Solve Equation 8 for z:

37z = 171 - 30y
z = (171 - 30y) / 37

8. Substitute the value of z in Equation 5:

-2x + 10y + 4((171 - 30y) / 37) = 50

Simplify Equation 5 and solve for x:

-2x + 10y + (684 - 120y) / 37 = 50
-2x + 370y + 684 - 120y = 1850
-2x - 120y = 1166
-2x = 1166 + 120y
x = (1166 + 120y) / -2
x = (583 - 60y) / -1

Therefore, the solution to the system of equations is:

x = (583 - 60y) / -1
y = y
z = (171 - 30y) / 37

To solve this system of equations, we can use the method of elimination or substitution. Let's use the method of elimination.

Step 1: Choose two equations and eliminate one variable.
We can eliminate the variable 'z' by multiplying the first equation by 2 and the second equation by 3:
(2 * (4x+2y-3z=7)) --> 8x + 4y - 6z = 14
(3 * (2x+4y+z=35)) --> 6x + 12y + 3z = 105

Step 2: Add the resulting equations together to eliminate the 'z' variable.
(8x + 4y - 6z) + (6x + 12y + 3z) = 14 + 105
Simplifying, we get:
14x + 16y = 119

Step 3: Choose another pair of equations and eliminate the same variable.
We can eliminate the variable 'z' again by multiplying the first equation by 3 and the third equation by 1:
(3 * (4x+2y-3z=7)) --> 12x + 6y - 9z = 21
(1 * (6x+2y-2z=20)) --> 6x + 2y - 2z = 20

Step 4: Subtract the resulting equations to eliminate the 'z' variable.
(12x + 6y - 9z) - (6x + 2y - 2z) = 21 - 20
Simplifying, we get:
6x + 4y - 7z = 1

Step 5: Now we have a system of two equations:
14x + 16y = 119
6x + 4y - 7z = 1

Step 6: Solve the system of equations.
To solve the system of equations, we need another equation that includes the variables 'x' and 'y'. Let's go back to one of the original equations:

2x + 4y + z = 35

We can solve this equation for 'z':
z = 35 - 2x - 4y

Now, substitute this value of 'z' into the second equation of the system:
6x + 4y - 7(35 - 2x - 4y) = 1

Simplify and solve for 'x':
6x + 4y - 245 + 14x + 28y = 1
20x + 32y = 246

Divide the equation by 4 to simplify it further:
5x + 8y = 61

Now, we have a system of two equations:
14x + 16y = 119
5x + 8y = 61

We can solve this system by multiplying the second equation by 2:
10x + 16y = 122

Now, subtract this equation from the first equation:
(14x + 16y) - (10x + 16y) = 119 - 122
4x = -3

Finally, solve for 'x':
x = -3/4 or -0.75

Substitute the value of 'x' into the second equation to solve for 'y':
5(-3/4) + 8y = 61
-15/4 + 8y = 61
8y = 61 + 15/4
8y = 244/4 + 15/4
8y = 259/4
y = (259/4) / 8
y = 259/32 or 8.09

Now that we have the values of 'x' and 'y', we can substitute them into any of the original equations to find 'z'. Let's use the first equation:
4(-3/4) + 2(8.09) - 3z = 7
-3 + 16.18 - 3z = 7
13.18 - 3z = 7
-3z = 7 - 13.18
-3z = -6.18
z = -6.18 / -3
z = 2.06

So, the solution to the system of equations is x = -0.75, y = 8.09, and z = 2.06.