A coffee distributor wants to mix an amount of House coffee that sells for 13.40 a pound with some Mexican Shade Grown coffee that sells for 11.20 a pound to create 55 pounds of a new coffee blend that will be priced at 12.28 per pound. How many pounds of each kind of coffee should they blend together? Round your answers to the nearest whole number of pounds. Enter your answer \( \begin{array}{ll}\sqrt{x} & \text { pounds of the House coffee } \\ \text { Enter your answer } & \text { I } \\ \sqrt{x} & \text { pounds of the Mexican Shade Grown coffee. }\end{array} \)

To solve this problem, we can set up a system of equations based on the given information. Let's denote the amount of House coffee as \( x \) pounds and the amount of Mexican Shade Grown coffee as \( y \) pounds. Given: 1. The total weight of the new coffee blend is 55 pounds: \( x + y = 55 \) 2. The price of the new coffee blend is 12.28 per pound: \( 12.28(x + y) = 13.40x + 11.20y \) We can solve this system of equations to find the values of \( x \) and \( y \), which represent the pounds of House coffee and Mexican Shade Grown coffee, respectively. Let's solve the system of equations. Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}x+y=55\\12.28\left(x+y\right)=13.4x+11.2y\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=55-y\\12.28\left(x+y\right)=13.4x+11.2y\end{array}\right.\) - step2: Substitute the value of \(x:\) \(12.28\left(55-y+y\right)=13.4\left(55-y\right)+11.2y\) - step3: Simplify: \(675.4=737-2.2y\) - step4: Swap the sides: \(737-2.2y=675.4\) - step5: Move the constant to the right side: \(-2.2y=675.4-737\) - step6: Subtract the numbers: \(-2.2y=-61.6\) - step7: Change the signs: \(2.2y=61.6\) - step8: Divide both sides: \(\frac{2.2y}{2.2}=\frac{61.6}{2.2}\) - step9: Divide the numbers: \(y=28\) - step10: Substitute the value of \(y:\) \(x=55-28\) - step11: Calculate: \(x=27\) - step12: Calculate: \(\left\{ \begin{array}{l}x=27\\y=28\end{array}\right.\) - step13: Check the solution: \(\left\{ \begin{array}{l}x=27\\y=28\end{array}\right.\) - step14: Rewrite: \(\left(x,y\right) = \left(27,28\right)\) The solution to the system of equations is \( x = 27 \) pounds of House coffee and \( y = 28 \) pounds of Mexican Shade Grown coffee. Therefore, the coffee distributor should blend 27 pounds of House coffee and 28 pounds of Mexican Shade Grown coffee to create 55 pounds of the new coffee blend.