2. Chapter 7: Systems of Equations and Inequalities; Matrices
7.2 Solution of Linear Systems in Three Variables
7.3 Solution of Linear Systems by Row Transformations
7.4 Matrix Properties and Operations
7.5 Determinants and Cramer’s Rule
7.6 Solution of Linear Systems by Matrix Inverses
7.7 Systems of Inequalities and Linear Programming
7.8 Partial Fractions

3. 7.2 Solution of Linear Systems in Three Variables
Solutions of systems with 3 variables with linear equations of the form
Ax + By + Cz = D (a plane in 3-D space)
are called ordered triples (x, y, z).
Possible solutions:

4. 7.2 Solving a System of Three Equations in Three Variables
Solve the system.
Eliminate z by adding (2) and (3)
3x + 2y = 4 (4)
Eliminate z from another pair of equations, multiply (2) by 6 and add the result to (1).
Eliminate x from equations 4 and 5.
Multiply (4) by -5 and add to (5).

5. 7.2 Solving a System of Three Equations in Three Variables
continued
Using y = –1, find x from equation (4) by substitution.
3x + 2(–1) = 4
x = 2
Substitute 2 for x and –1 for y in equation (3) to find z.
2 + (–1) + z = 2
z = 1
The solution set is {(2, –1, 1)}.

6. 7.2 Solving a System of Two Equations and Three Unknowns
Example Solve the system.
Solution Geometrically, the solution of two non-
parallel planes is a line. Thus, there will be an infinite
number of ordered triples in the solution set.

7. 7.2 Solving a System of Two Equations and Three Unknowns
This is as far as we can go with the echelon method.
Solve y + z = 3 to get y = 3 – z for any arbitrary
value for z. Now we express x in terms of z by
solving equation (1).

8. 7.2 Solving a System of Two Equations and Three Unknowns
The solution set is written {(z – 2, 3 – z, z)}. For
example, if z = 1, then y = 3 – 1 = 2 and
x = 1 – 2 = –1, giving the solution set {(–1, 2, 1)}.
Verify that another solution is {(0, 1, 2)}.
Let y = 3 – z.

9. 7. 2. Application: Solving a System of Three
7.2 Application: Solving a System of Three Equations to Satisfy Feed Requirements
An animal feed is made from three ingredients: corn, soybeans,
and cottonseed. One unit of each ingredient provides units of
protein, fat, and fiber, as shown in the table. How many units
of each ingredient should be used to make a feed that contains
22 units of protein, 28 units of fat, and 18 units of fiber?

10. 7. 2. Application: Solving a System of Three
7.2 Application: Solving a System of Three Equations to Satisfy Feed Requirements
Solution Let x represent the number of units of corn,
y, the number of units of soybeans, and z, the number
of units of cottonseed. Using the table, we get the following
system
or,
We can show that x = 40, y = 15, and z = 30.

11. 7.2 Curve Fitting Using A System
Find the equation of the parabola with vertical axis that passes through the points (2, 4), (1, 1), and (2, 5).
Substitute each ordered pair into the equation ax2 + bx + c.
4 = 4a + 2b + c (1)
1 = a – b + c (2)
5 = 4a – 2b + c (3)
Eliminate c using equations (1) and (2).
Eliminate c using equations (2) and (3).

12. 7.2 Curve Fitting Using A System
continued
Solve the system of equations in two
variables by eliminating a.
Find a by substituting for Find c by substituting
b in equation (4) for a and b in equation (2)
Find c by substituting a and b in equation (2).
The equation of the parabola is: