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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