Find the Variables

Problem

[[1,1,0,0],[0,0,1,1],[1,0,1,0],[0,1,0,1]]*[[x],[y],[v],[z]]=[[110],[160],[140],[130]]

Solution

1. Write the system of linear equations by performing matrix multiplication.

x+y=110

v+z=160

x+v=140

y+z=130

2. Express y in terms of x using the first equation.

y=110−x

3. Express v in terms of x using the third equation.

v=140−x

4. Substitute the expressions for y and v into the remaining equations to check for consistency and solve for z

(140−x)+z=160⇒z=x+20

(110−x)+z=130⇒z=x+20

5. Observe that the system is dependent because the fourth equation yields the same relationship as the others; let x be a free parameter k

x=k

y=110−k

v=140−k

z=k+20

6. Solve for a specific solution if one is implied, or provide the general solution; assuming a standard algebraic context where a unique solution might be expected, we check the sum of equations.

(x+y)+(v+z)=110+160=270

(x+v)+(y+z)=140+130=270

7. Identify that without further constraints, the variables are defined in terms of x

x=x

y=110−x

v=140−x

z=x+20

Final Answer

[[x],[y],[v],[z]]=[[x],[110−x],[140−x],[x+20]]

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