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Section 4-6 Matrix Equations and Systems of Linear Equations
Example 1
2 x1  2 x2  x3  3
3x1  x2  x3  7
x1  3x 2  2 x3  0
A system of n linear equations in n unknowns can be written as a matrix equation of the form
AX = B where A is the square matrix of coefficients, X is the column matrix of variables, and B is
the column matrix of constants. For example, our system of linear equations can be written as
the matrix equation
2  2 1   x1  3
3 1  1  x   7

 2   
1  3 2   x3  0
Given the matrix equation AX = B, multiply both sides on the left by A-1:
A1 AX  A1 B  IX  A1 B  X  A1 B
If A-1 doesn’t exist then either there is no solution or there are multiple solutions.
To solve the system of equations given above, enter matrix A (the coefficients) and matrix B
(the constants) into your TI-83 calculator. The expression A-1B will generate the corresponding x
values as a column matrix:
 x1   2 
x    0 
 2  
 x3   1
Example 2
x1 -
x2 + x3 = 3
2x2 – x3 = 1
2x1 + 3x2
= 4
1
1  1 1   x1  3
 x1  1  1 1  3  8 
0 2  1  x   1   x   0 2  1 1   4

 2   
 2 
    
2 3 0   x3  4
 x3  2 3 0  4   9
Example 4
Labor and material costs for manufacturing two guitar models are given in the table:
Guitar Model Labor Cost Material Costs
A
$30
$20
B
$40
$30
(A) If a total of $3,000 is allowed for labor and material, how many of each model should be
produced each week to use exactly each of the allocations of the $3,000 indicated in the
following table?
Weekly Allocation
1
2
3
Labor
$1,800 $1,750 $1,720
Material $1,200 $1,250 $1,280
Weekly Allocation #1
Model #
A
x1
B
x2
Model A
Model B Constraint
Labor
30x1
+
40x2
= 1800
Material
20x1
+
30x2
= 1200
1
30 40 1800 60
X 
 
     x1  60 Model A and x2  0 Model B
20 30 1200  0 
The red line represents the labor constraint and the green line represents the material
constraint. They intersect at (60, 0).
Weekly Allocation #2
Model #
A
x1
B
x2
Model A
Model B Constraint
Labor
30x1
+
40x2
= 1750
Material
20x1
+
30x2
= 1250
1
30 40 1750 25
X 
 
     x1  25 Model A and x2  25 Model B
20 30 1250 25
The red line represents the labor constraint and the green line represents the material
constraint. They intersect at (25, 25).
Weekly Allocation #3
Model #
A
x1
B
x2
Model A
Model B Constraint
Labor
30x1
+
40x2
= 1720
Material
20x1
+
30x2
= 1280
1
30 40 1720  4 
X 
 
     x1  4 Model A and x2  40 Model B
20 30 1280 40
The red line represents the labor constraint and the green line represents the material
constraint. They intersect at (4, 40).
(B) Is it possible to use an allocation of $1,600 for labor and $1,400 for material?
Model #
A
x1
B
x2
Model A
Model B Constraint
Labor
30x1
+
40x2
= 1600
Material
20x1
+
30x2
= 1400
1
30 40 1600  80
X 
 

  There is no feasible solution
20 30 1400  100 
The red line represents the labor constraint and the green line represents the material
constraint. They do not intersect in the region in which both x1 and x2 are positive.
What about $2,000 for labor and $1,000 for material?
Model #
A
x1
B
x2
Model A
Model B Constraint
Labor
30x1
+
40x2
= 2000
Material
20x1
+
30x2
= 1000
1
30 40 2000  200 
X 
 

  There is no feasible solution
20 30 1000   100
The red line represents the labor constraint and the green line represents the material
constraint. They do not intersect in the region in which both x1 and x2 are positive.
Example 5
A corporation has a taxable income of $7,650,000. At this income level, the federal income tax rate is
50%, the state tax rate is 20%, and the local tax rate is 10%. If each tax rate is applied to the total taxable
income, the resulting tax liability for the corporation would be 80% of taxable income. However, it is
customary to deduct taxes paid to one agency before computing taxes for the other agencies. Assume
that the federal taxes are based on the income that remains after the state and local taxes are
deducted, and that the state and local taxes are computed in a similar manner. What is the tax liability
of the corporation (as a percentage of taxable income) if these deductions are taken into consideration?
Agency Tax Liability
Fed
x1
State
x2
Local
x3
Agency
Fed
State
Local
Tax Liability
x1 = 0.5(7650000 – x2 – x3)
x2 = 0.2(7650000 – x1 – x3)
x3 = 0.1(7650000 – x1 – x2)
Each of these three equations can be put into standard form (variables on the left and constants on the
right):
Fed
x1 + 0.5x2 + 0.5x3 = 3825000
State 0.2x1 +
x2 + 0.2x3 = 1530000
Local 0.2x1 + 0.2x2 +
x3 = 765000
1
 1 0.5 0.5 3825000 $3,240,000 Federal taxes
0.2 1 0.2 1530000    $810,000  State taxes

 
 

 0.1 0.1 1   765000   $360,000  Local taxes
The total tax liability is $4,410,000 or
about 57.65% of the taxable income.
The graphical interpretation of this
problem is a little harder because we are
dealing with three dimensions. A linear
equation in three dimensions defines a
plane. The red plane represents the federal
tax equation, the green plane represents
the state tax equation and the purple
plane represents the local tax equation.
The intersection of two planes is a straight
line. The intersections of the three pairs of
equations are indicated as black lines and
labeled in the graph. The intersection of
these three lines represents the point at
which all three planes intersect and this
point represents the solution of the system
of equations (3.24, 0.81, and 0.36) in
millions of dollars.