Find the Variables

Problem

[[1,1,1],[4,10,0],[4√(,3),0,−2]]*[[x],[y],[z]]=[[600],[4500],[0]]

Solution

1. Write the system of linear equations from the matrix multiplication.

x+y+z=600

4*x+10*y=4500

4√(,3)*x−2*z=0

2. Solve the third equation for z in terms of x

2*z=4√(,3)*x

z=2√(,3)*x

3. Solve the second equation for y in terms of x

10*y=4500−4*x

y=450−0.4*x

4. Substitute the expressions for y and z into the first equation.

x+(450−0.4*x)+(2√(,3)*x)=600

5. Combine like terms to solve for x

0.6*x+2√(,3)*x=150

x*(0.6+2√(,3))=150

x=150/(0.6+2√(,3))

6. Calculate the numerical value for x (using √(,3)≈1.732.

x≈150/(0.6+3.464)

x≈36.91

7. Substitute x back into the expressions for y and z

y=450−0.4*(36.91)≈435.24

z=2√(,3)*(36.91)≈127.85

Final Answer

[[x],[y],[z]]=[[150/(0.6+2√(,3))],[450−60/(0.6+2√(,3))],[(300√(,3))/(0.6+2√(,3))]]≈[[36.91],[435.24],[127.85]]

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