Solve the system of equations to find x,y,z (hint let d=x2 e=y2 and d=z2 . Find d e and f first) 2 means squared I don't have it on my keyboard.
X2+y2+z2=9
3x2-y2-z2=7
Y2+2z2 =6
x^2 + y^2 + z^2 = 9 sphere center at origin radius 3
or using that suggested notation
d + e + f = 9
3d - e -f = 7
e + 2f = 6
3d+3e+3f = 27
3d-e-f=7
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4e+4f=20
e+f = 5
e+2f=6
e+1f=5
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f=1
then e = 4
d+1+4=9
d =4
so
x=2
y=2
z=1
Thank you
To solve the system of equations and find the values of x, y, and z, we'll follow these steps:
Step 1: Let's define d = x^2, e = y^2, and f = z^2. This substitution will help simplify the equations.
Step 2: Rewrite the system of equations using the new variables:
d + e + f = 9 ...(Equation 1)
3d - e - f = 7 ...(Equation 2)
e + 2f = 6 ...(Equation 3)
Step 3: We have three equations (Equations 1, 2, and 3) with three variables (d, e, and f). We can solve this system of equations using various methods. Let's use the elimination method:
Multiply Equation 3 by 3:
3e + 6f = 18 ...(Equation 4)
Now, we can add Equation 4 to Equation 2 to eliminate the variable e:
3d - e - f + 3e + 6f = 7 + 18
3d + 5f = 25 ...(Equation 5)
Step 4: Now we have two equations (Equations 1 and 5) with variables d and f. Let's solve this system of equations using substitution:
Rearrange Equation 5 to solve for d:
3d = 25 - 5f
d = (25 - 5f)/3 ...(Equation 6)
Substitute the expression for d from Equation 6 into Equation 1:
(25 - 5f)/3 + e + f = 9
Multiply through by 3 to eliminate the fraction:
25 - 5f + 3e + 3f = 27
3e - 2f = 2 ...(Equation 7)
Step 5: Now, we have two equations (Equations 6 and 7) with variables e and f. Let's solve this system of equations using elimination:
Multiply Equation 7 by 3:
9e - 6f = 6 ...(Equation 8)
Add Equation 8 to (2 * Equation 6):
9e - 6f + 10f - 6f = 6 + 50/3
9e - 2f = 98/3
Divide through by 3 to simplify:
3e - 2f = 98/9 ...(Equation 9)
Step 6: We now have two equations (Equations 9 and 7) with variables e and f. Let's solve this system of equations using substitution:
Rearrange Equation 9 to solve for e:
3e = 98/9 + 2f
e = (98/9 + 2f)/3 ...(Equation 10)
Substitute the expression for e from Equation 10 into Equation 7:
3((98/9 + 2f)/3) - 2f = 2
Simplifying:
98/9 + 2f - 2f = 2
98/9 = 2
This is a contradiction, and the system of equations is inconsistent. Hence, there is no solution for x, y, and z that satisfies all three equations.