1. Systems of Equations and Inequalities
Chapter 4
Systems of Equations and Inequalities

2. Chapter Sections
4.1 – Solving Systems of Linear Equations in Two Variables
4.2 – Solving Systems of Linear Equations in Three Variables
4.3 – Systems of Linear Equations: Applications and Problem Solving
4.4 – Solving Systems of Equations Using Matrices
4.5 – Solving Systems of Equations Using Determinants and Cramer’s Rule
4.6 – Solving Systems of Linear Inequalities
Chapter 1 Outline

3. Solving Systems of Linear Equations in Three Variables
§ 4.2
Solving Systems of Linear Equations in Three Variables

4. Definitions
The equation 2x – 3y + 4z = 8 is an example of a linear equation in three variables. The solution to this type of equation is an ordered triple of the form (x, y, z).
One possible solution to the equation
5x – 3y + 4z = 9 is
(1, 2, 3).

5. Solving Systems
Systems in three (or more) linear equations are solved the same way systems of two linear equations are solved by using either the substitution or addition method.
Solve the following system of equations using the substitution method.

6. Solving Systems
Since we know that x = -3, we can substitute it into the equation 3x + 4y = 7 and solve for y.

7. Solving Systems
Now we substitute x=-3 and y=4 into the last equation and solve for z.
The solution is the ordered triple (-3, 4, 5).

8. Geometric Interpretation
The following is a geometric interpretation of the solution (4, 5, 3). x y z
(4, 5, 3) 4 3 5

9. Inconsistent and Dependent Systems
Inconsistent System of Equations
A system that has no solution. Example: You obtain a statement that is always false, such as 3=0.
Dependent System of Equations
A system that has an infinite number of solutions. Example: You obtain a statement that is always true, such as 0=0.