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Multiply one or both equations by some value to create equal or opposite coefficients. Then use the Elimination Method to solve the system.
Nathan buys 2 cups of hot chocolate and 5 cups of coffee.
We are told that Nathan pays a total of $ 11.15 for 7 cups of coffee and hot chocolate. Each cup of coffee costs $ 1.75, and each cup of hot chocolate $ 1.20. We want to know how many of each he bought. To answer this question, we will set up and solve a system of linear equations. Let's define the variables first.
Let's now write the verbal phrases as algebraic phrases.
| Verbal phrase | Algebraic phrase |
|---|---|
| The number of cups of coffee plus the number of cups of hot chocolate is 7 |
c + h = 7 |
| The number of cups of coffee plus the number of cups of hot chocolate have a total cost of $ 11.15 |
1.75c + 1.20h = 11.15 |
Therefore, our system of equations can be written as follows. c+h=7 & (I) 1.75c+1.20h=11.15 & (II) Note that neither variable has equal or opposite coefficients in the equations. This means we first need to multiply one or both equations by some value to create equal or opposite coefficients. Then we can solve using the Elimination Method. (I): c+h=7 * 1.20 1.20c+1.20h=8.4 Therefore, our system now is as follows. 1.20c+1.20h=8.4 & (I) 1.75c+1.20h=11.15 & (II) To solve using the Elimination Method, we will start by subtracting Equation (I) from Equation (II).
(II): Subtract (I)
We have found that c=5. To find the value of h, we will substitute 5 for c in Equation (I) and solve for h.
We have found that h=2. Therefore, the solution of the system, which is the point of intersection of the lines, is (5,2). In the context of the problem, this means that Nathan bought 2 cups of hot chocolate and 5 cups of coffee.