Question

Three market leaders in homeowners insurance in a state are State Farm, American Family Insurance Group, and Allstate. Based on data from a certain year, two relationships between the homeowners insurance percentage market shares are found to be as given below, where x, y, z, and w are the percentages of the market held by State Farm, American Family, Allstate, and other companies, respectively.

x = −56 + 5y + z

z = 7 − 2y + w

Given that the four groups account for the entire market, obtain a third equation relating x, y, z, and w, and solve the associated system of three linear equations to show how the market shares of State Farm, American Family, and Allstate depend on the share held by other companies. (Express your answer in terms of w, where x = x(w), y = y(w),and z = z(w).)

(x, y, z)= ?

Answer 1

Given that the four groups account for the entire market, then a third equation relating x, y, z, and w, is given by:

$x+y+z+w=100$

So, the system of linear equations is given by:

$\left\{\begin{matrix} x-5y-z+0w=-56\\ 0x+2y+z-w=7\\ x+y+z+w=100 \end{matrix}\right.$

which is a system with three equations and four unknowns and where its matrix is:

/1-5-1 0-56 0 21 7 100

It is solve with Gauss-Jordan elimination and it get:

Step 1: Subtract row 1 from row 3

/1-5-1 0-56 0 21 7 156

Step 2: Swap row 3 and 2

/1-5-1 0-56 0 6 21 156 0 21 7

Step 3: Divide row 2 by 6

/1-5-1 0-56 126 0 21 7

Step 4: Subtract (2 × row 2) from row 3

/1-5-1 0-56 1 26 45

Step 5: Multiply row 3 by 3

/1-5-1 0-56 3626 0 0 1 -4 -135

Step 6: Subtract (-1 × row 3) from row 1

$\begin{pmatrix} 1 & -5 & 0& -4& -191\\ 0 & 1 & \frac{1}{3}&\frac{1}{6} &26 \\ 0& 0 & 1 & -4 & -135 \end{pmatrix}$

Step 7: Subtract (0.333 × row 3) from row 2

く1-50-4-191 3 71 0 0 1 -4 -135

Step 8: Subtract (-5 × row 2) from row 1

$\begin{pmatrix} 1 & 0 & 0& \frac{7}{2}& 164\\ 0 & 1 & 0&\frac{3}{2} &71 \\ 0& 0 & 1 & -4 & -135 \end{pmatrix}$

Matrix is now in reduced row echelon form.

Therefore,

$x=x(w)=164-\frac{7}{2}w$

  $y=y(w)=71-\frac{3}{2}w$

  $z=z(w)=-135+4w$

where w can take any value.

or, what is equivalent,

$(x,y,z)=\left ( 164-\frac{7}{2}w,71-\frac{3}{2}w,-135+4w \right )$

where w can take any value.