Elementary Linear Algebra 7th Edition

Published by Cengage Learning
ISBN 10: 1-13311-087-8
ISBN 13: 978-1-13311-087-3

Chapter 1 - Systems of Linear Equations - 1.1 Introduction to Systems of Linear Equations - 1.1 Exercises: 65

Answer

One eight-ounce glass of apple juice contains $95.5$ milligrams of vitamin C. One eight-ounce glass of orange juice contains $81.9$ milligrams of vitamin C.

Work Step by Step

We will solve this problem by writing the appropriate system of equations and then solving it. Let $x$ denote the amount of vitamin C in one eight-ounce glass of apple juice and $y$ the amount of vitamin C in one eight-ounce glass of orange juice in milligrams. Since one glass of apple and one glass of orange juice add up to $177.4$ milligrams the first equation is: $$x+y=177.4.$$ Since two glasses of apple and three glasses of orange juice contain $436.7$ milligrams the second equation is: $$2x+3y=436.7.$$ So now we have \begin{align*} x+y=&177.4\\ 2x+3y=&436.7 \end{align*} Now let's solve this system. Step 1: Express $y$ from the first equation: $$y=177.4-x$$ Step 2: Put this into the second equation and solve for $x$: $$2x+3(177.4-x) = 436.7$$ which gives $$-x+532.2=436.7$$ and that at the end gives $$x=532.2-436.7=95.5$$ Step 3: Use the calculated value for $x$ to solve for $y$: $$y=177.4-x=177.4-95.5=81.9$$ So one eight-ounce glass of apple juice contains $95.5$ milligrams of vitamin C and one eight-ounce glass of orange juice contains $81.9$ milligrams of vitamin C.
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