SECTION 8.1
Matrix Solutions to Linear Systems
Using Matrices to Solve Linear Systems
• We can use matrices, applying Gaussian Elimination, to
solve linear systems.
Example 8
The nutritional content per ounce of three foods is
presented in the table on the right. If a meal consisting of
the three foods allows exactly 2200 calories, 110 grams of
protein, and 900 milligrams of vitamin C, how many ounces
of each kind of food should be used?
Calories Protein
Vitamin C
(in grams) (in milligrams)
Food A
100
9
50
Food B
400
8
250
Food C
300
15
100
SECTION 8.2
Inconsistent and Dependent Systems
Possible Solutions to
Linear Systems in 3 Variables
• There are three ways that we can arrive at no solution.
Example 1
• Use Gaussian elimination to find the complete solution or
show that none exists.
2𝑥 − 4𝑦 + 𝑧 = 3
𝑥 − 3𝑦 + 𝑧 = 5
3𝑥 − 7𝑦 + 2𝑧 = 12
Example 2
• Use a calculator to find the complete solution or show that
none exists.
5𝑥 + 12𝑦 + 𝑧 = 10
2𝑥 + 5𝑦 + 2𝑧 = −1
𝑥 + 2𝑦 − 3𝑧 = 5
Possible Solutions to
Linear Systems in 3 Variables
• There are two ways that we can arrive at infinitely many
solutions.
Example 3
• Use Gaussian elimination to find the complete solution or
show that none exists
5𝑥 − 11𝑦 + 6𝑧 = 12
−𝑥 + 3𝑦 − 2𝑧 = −4
3𝑥 − 5𝑦 + 2𝑧 = 4
Example 4
• Use a calculator to find the complete solution or show that
none exists.
8𝑥 + 5𝑦 + 11𝑧 = 30
−𝑥 − 4𝑦 + 2𝑧 = 3
2𝑥 − 𝑦 + 5𝑧 = 12
Example 5
• Use Gaussian elimination to find the complete solution or
show that none exists
3𝑥 + 7𝑦 + 6𝑧 = 26
𝑥 + 2𝑦 + 𝑧 = 8