-.30x+.20y +.25z =0
.20x -.40y +.05z =0
.10x +.20y -.30z =0
x=?
y=?
z=?
To find the values of x, y, and z in the given system of equations, you can use the method of elimination or substitution. Let's use the method of elimination here.
Step 1: Multiply the first equation by 10, the second equation by 5, and the third equation by 20 to eliminate the decimal coefficients.
Multiply the first equation by 10:
-3x + 2y + 2.5z = 0
Multiply the second equation by 5:
x - 2y + 0.25z = 0
Multiply the third equation by 20:
2x + 4y - 6z = 0
The system of equations becomes:
-3x + 2y + 2.5z = 0
x - 2y + 0.25z = 0
2x + 4y - 6z = 0
Step 2: Add the first equation to the second equation to eliminate y.
-3x + 2y + 2.5z + x - 2y + 0.25z = 0
Combine like terms:
-2x + 2.75z = 0
The system becomes:
-2x + 2.75z = 0
x - 2y + 0.25z = 0
2x + 4y - 6z = 0
Step 3: Add 2 times the first equation to the third equation to eliminate x.
-2x + 2.75z + 2(2x + 4y - 6z) = 0
Simplify and combine like terms:
2.75z - 2x + 4x + 8y - 12z = 0
2x + 8y - 9.25z = 0
The system becomes:
-2x + 2.75z = 0
x - 2y + 0.25z = 0
2x + 8y - 9.25z = 0
Step 4: Multiply the first equation by 2 and add it to the second equation to eliminate x.
2(-2x + 2.75z) + (x - 2y + 0.25z) = 0
-4x + 5.5z + x - 2y + 0.25z = 0
Combine like terms:
-3x - 2y + 5.75z = 0
The system becomes:
-3x - 2y + 5.75z = 0
x - 2y + 0.25z = 0
2x + 8y - 9.25z = 0
Step 5: Now we have a system of three equations:
-3x - 2y + 5.75z = 0
x - 2y + 0.25z = 0
2x + 8y - 9.25z = 0
We can solve this system using any method, such as substitution or matrix operations. Let's use substitution to find the values of x, y, and z:
From the second equation, solve for x:
x = 2y - 0.25z
Substitute this value of x into the first equation:
-3(2y - 0.25z) - 2y + 5.75z = 0
-6y + 0.75z - 2y + 5.75z = 0
Combine like terms:
-8y + 6.5z = 0
Now substitute the value of x into the third equation:
2(2y - 0.25z) + 8y - 9.25z = 0
4y - 0.5z + 8y - 9.25z = 0
Combine like terms:
12y - 9.75z = 0
Now we have a system of two equations:
-8y + 6.5z = 0
12y - 9.75z = 0
We can solve this system using substitution or elimination again:
Let's solve for y in terms of z from the first equation:
-8y + 6.5z = 0
-8y = -6.5z
y = (6.5/8)z
y = 0.8125z
Substitute this value of y into the second equation:
12(0.8125z) - 9.75z = 0
9.75z - 9.75z = 0
0 = 0
Since we get an identity equation (0 = 0), it means that the system of equations is dependent and has infinitely many solutions.