# Solve the system of equations using matrices? # 16x + 5y = 211 # and # 16x + y = 183 #

##
(Question Restore: portions of this question have been edited or deleted!)

(Question Restore: portions of this question have been edited or deleted!)

##### 1 Answer

(B)

#### Explanation:

We have:

# 16x + 5y = 211 #

# 16x + y = 183 #

Which we can write in vector matrix form:

# ( (16,5), (16,1) ) ( (x), (y) ) = ( (211), (183) ) #

So, pre-multiplying by the inverse matrix we have:

# ( (16,5), (16,1) )^(-1)( (16,5), (16,1) ) ( (x), (y) ) = ( (16,5), (16,1) )^(-1)( (211), (183) ) #

# :. ( (x), (y) ) = ( (16,5), (16,1) )^(-1)( (211), (183) ) #

Or:

# bb(A) bb(ul x) = bb(ul b) => bb(ul x) = bb(A)^(-1) bb(ul b) #

Where

# bb(A) = ( (16,5), (16,1) ) # ;# bb(ul x) = ( (x), (y) ) # ;# bb(ul b) ( (211), (183) ) #

We can find

A matrix,

- Calculating the Matrix of Minors,
- Form the Matrix of Cofactors,
#cof(bb(A))# - Form the adjoint matrix,
#adj(bb(A))# - Multiply
#adj(bb(A))# by#1/abs(bb(A))# to form the inverse#bb(A)^-1#

At some point we need to calculate

# bb(A) = ( (16,5), (16,1)) #

If we expand about the first row;

# abs(bb(A)) = (15)(1) - (16)(5) #

# \ \ \ \ \ = 16-80 #

# \ \ \ \ \ = -64 #

As

#"minors"(bb(A)) = ( (1, 16), (5, 16 ))#

We now form the matrix of cofactors,

# ( (+, -), (-, +) )#

Where we change the sign of those elements with the minus sign to get;

# cof(bbA) = ( (1, -16), (-5, 16 )) #

Then we form the adjoint matrix by transposing the matrix of cofactors,

#adj(A) = cof(A)^T#

#\ \ \ \ \ \ \ \ \ \ \ = ( (1, -16), (-5, 16 ))^T #

#\ \ \ \ \ \ \ \ \ \ \ = ( (1, -5), (-16, 16 )) #

And then finally we multiply by the reciprocal of the determinant to get:

#bb(A)^-1 = 1/abs(bb A) adj(bb A)#

#\ \ \ \ \ \ \ = 1/(-64) ( (1, -5), (-16, 16 )) #

So then we get the solution the linear equations as:

# bb(ul x) = bb(A)^(-1) bb(ul b) # .....#[star]#

# :. ( (x), (y) ) = 1/(-64)( (1, -5), (-16, 16 )) ( (211), (183) ) #

# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = -1/64( ((1)(211)+(-5)(183) ), ((-16)(211)+(16)(183) ) ) #

# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = -1/64( (-704), (-448) ) #

# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = ( (11), (7) ) #

Hence the solution equation is:

# ( (x), (y) ) = ( (16,5), (16,1) )^(-1)( (211), (183) ) = ( (11), (7) )#

Making (B) the coirerct solution