Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Westminster City Cafe recently introduced a new flavor of coffee. They served $65$ grande cups and $30$ jumbo cups of the new coffee today, which equaled a total of $41,175$ grams. The day before, $65$ grande cups and $31$ jumbo cups were served, which used a total of $41,709$ grams. How much coffee is required to make each size? $\newline$ There are $\_$ grams in a grande cup of coffee and $\_$ grams in a jumbo one.

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Solve a system of equations using any method: word problems

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

\newline

Westminster City Cafe recently introduced a new flavor of coffee. They served

65

grande cups and

30

jumbo cups of the new coffee today, which equaled a total of

41,175

grams. The day before,

65

grande cups and

31

jumbo cups were served, which used a total of

41,709

grams. How much coffee is required to make each size?

\newline

There are

\_

grams in a grande cup of coffee and

\_

grams in a jumbo one.

Equation $1$ : Let's denote the amount of coffee required for a grande cup as ' $g$ ' grams and for a jumbo cup as ' $j$ ' grams. $\newline$ The first equation comes from the first day's servings: $65$ grande cups and $30$ jumbo cups made a total of $41,175$ grams. $\newline$ So, the equation is: $65g + 30j = 41,175$ .
Equation $2$ : The second equation comes from the second day's servings: $65$ grande cups and $31$ jumbo cups made a total of $41,709$ grams. $\newline$ So, the equation is: $65g + 31j = 41,709$ .
System of Equations: We now have a system of equations: $\newline$ $1$ ) $65g + 30j = 41,175$ $\newline$ $2$ ) $65g + 31j = 41,709$ $\newline$ We can solve this system by subtracting the first equation from the second to eliminate ' $g$ '.
Subtracting Equations: Subtracting the first equation from the second gives us: $\newline$ $(65g + 31j) - (65g + 30j) = 41,709 - 41,175$ $\newline$ This simplifies to: $\newline$ $65g + 31j - 65g - 30j = 41,709 - 41,175$ $\newline$ Which further simplifies to: $\newline$ $j = 534$
Solving for j: Now that we have the value for 'j', we can substitute it back into one of the original equations to solve for 'g'. $\newline$ Let's use the first equation: $65g + 30j = 41,175$ . $\newline$ Substituting 'j' with $534$ gives us: $\newline$ $65g + 30(534) = 41,175$ .
Substitute and Solve for g: Now we solve for 'g': $\newline$ $65g + 16,020 = 41,175$ $\newline$ $65g = 41,175 - 16,020$ $\newline$ $65g = 25,155$ $\newline$ $g = \frac{25,155}{65}$ $\newline$ $g = 387$