-.30x+.20y +.25z =0
.20x -.40y +.05z =0
.10x +.20y -.30z =0
x=?
y=?
z=?
no matter how you add/subtract or multiply/divide the left side of the equations, the right side remains zero
So clearly
x = 0
y = 0
z = 0
or each plane passes through the origin.
So they all intersect at the origin (0,0,0)
To find the values of x, y, and z that satisfy the given set of equations, we can solve them using the method of simultaneous equations. Here's how to do it step by step:
1. Rearrange the equations:
-.30x + .20y + .25z = 0 --> Equation 1
.20x - .40y + .05z = 0 --> Equation 2
.10x + .20y - .30z = 0 --> Equation 3
2. Choose two of the equations and eliminate one variable. Let's eliminate y:
Multiply Equation 1 by 2: -.60x + .40y + .50z = 0 --> Equation 4
Multiply Equation 2 by 1.5: .30x - .60y + .075z = 0 --> Equation 5
Add Equation 4 and Equation 5 to eliminate y:
(-.60x + .40y + .50z) + (.30x - .60y + .075z) = 0
-.30x + .575z = 0 --> Equation 6
3. Now, choose another pair of equations and eliminate another variable. Let's eliminate x:
Multiply Equation 2 by 3: .60x - 1.20y + .15z = 0 --> Equation 7
Multiply Equation 3 by 1: .10x + .20y - .30z = 0 --> Equation 8
Add Equation 7 and Equation 8 to eliminate x:
(.60x - 1.20y + .15z) + (.10x + .20y - .30z) = 0
.70y - .15z = 0 --> Equation 9
4. Now, we have Equations 6 and 9 with two variables. Solve these simultaneous equations to find the values of y and z.
Start by isolating y in Equation 9:
.70y = .15z
y = (.15z) / .70
Substitute this value of y into Equation 6:
-.30x + .575z = 0
-.30x = -.575z
x = (-.575z) / (-.30)
So, x = 1.917z and y = (0.15z) / 0.70
5. To determine the specific values of x, y, and z, you'll need additional information or constraints provided in the problem. Without additional information, the variables can take any numerical value as long as they satisfy the given equations.
Therefore, the solutions for x, y, and z are:
x = 1.917z
y = (0.15z) / 0.70
z = z, where z is any numerical value.