Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Students in a health class are tracking how much water they consume each day. Alexandra has two reusable water bottles: a small one and a large one. Yesterday, she drank $2$ small bottles and $2$ large bottles, for a total of $64$ ounces. The day before, she drank $1$ small bottle and $2$ large bottles, for a total of $51$ ounces. How much does each bottle hold? $\newline$ The small bottle holds _ ounces and the large one holds _ ounces.

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Solve a system of equations using elimination: word problems

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.

\newline

Students in a health class are tracking how much water they consume each day. Alexandra has two reusable water bottles: a small one and a large one. Yesterday, she drank

2

small bottles and

2

large bottles, for a total of

64

ounces. The day before, she drank

1

small bottle and

2

large bottles, for a total of

51

ounces. How much does each bottle hold?

\newline

The small bottle holds _____ ounces and the large one holds _____ ounces.

Define Variables: Let's define two variables: let $x$ be the number of ounces the small bottle holds, and $y$ be the number of ounces the large bottle holds. We can then write two equations based on the information given: $\newline$ $1$ . For the first day: $2x + 2y = 64$ (since Alexandra drank $2$ small and $2$ large bottles) $\newline$ $2$ . For the second day: $x + 2y = 51$ (since she drank $1$ small and $2$ large bottles)
Write Equations: To solve this system using elimination, we can multiply the second equation by $2$ to align the coefficients of $y$ : $\newline$ $2(x + 2y) = 2(51)$ $\newline$ This gives us a new equation: $\newline$ $2x + 4y = 102$
Multiply Second Equation: Now we have two equations with the same coefficient for $y$ : $1. 2x + 2y = 64$ $2. 2x + 4y = 102$ We can subtract the first equation from the second to eliminate $x$ : $(2x + 4y) - (2x + 2y) = 102 - 64$ This simplifies to: $2y = 38$
Eliminate $x$ : Divide both sides of the equation by $2$ to solve for $y$ : $\frac{2y}{2} = \frac{38}{2}$ $y = 19$ So, the large bottle holds $19$ ounces.
Solve for y: Now that we know the value of $y$ , we can substitute it back into one of the original equations to solve for $x$ . Let's use the second equation: $\newline$ $x + 2(19) = 51$ $\newline$ $x + 38 = 51$
Substitute Back: Subtract $38$ from both sides to solve for $x$ : $\newline$ $x + 38 - 38 = 51 - 38$ $\newline$ $x = 13$ $\newline$ So, the small bottle holds $13$ ounces.