Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. Gymnasts training for an upcoming competition are practicing their routines for balance beam and floor exercise. During morning practice, Quinn practiced her beam routine 3 times and her floor routine 1 time, which took a total of 6 minutes. During afternoon practice, she ran through her her beam routine 2 times and her floor routine 2 times, which took a total of 8 minutes. How long is each routine? The beam routine is _____ minutes long and the floor exercise is _____ minutes long.

Question

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.  Gymnasts training for an upcoming competition are practicing their routines for balance beam and floor exercise. During morning practice, Quinn practiced her beam routine 3 times and her floor routine 1 time, which took a total of 6 minutes. During afternoon practice, she ran through her her beam routine 2 times and her floor routine 2 times, which took a total of 8 minutes. How long is each routine?  The beam routine is _____ minutes long and the floor exercise is _____ minutes long.

Bytelearn · Accepted Answer

Define Variables: Let's define the variables for the lengths of the routines. Let x be the length of the beam routine in minutes, and y be the length of the floor routine in minutes.First Equation: According to the morning practice information, we can write the first equation based on the total time spent: 3x (beam routines) + 1y (floor routine) = 6 minutes. 3x + y = 6Second Equation: From the afternoon practice information, we can write the second equation: 2x (beam routines) + 2y (floor routines) = 8 minutes. 2x + 2y = 8System of Equations: We now have a system of equations: 3x + y = 6 2x + 2y = 8 We can solve this system using either substitution or elimination. Let's use the elimination method.Elimination Method: To eliminate y, we can multiply the first equation by 2, so the coefficients of y in both equations match. (3x + y) * 2 = 6 * 2 6x + 2y = 12Eliminate y: Now we subtract the second equation from the modified first equation to eliminate y: (6x + 2y) - (2x + 2y) = 12 - 8 6x + 2y - 2x - 2y = 12 - 8 4x = 4Subtract Equations: Solve for x: 4x = 4 x = 4 / 4 x = 1 So, the beam routine is 1 minute long.Solve for x: Now we substitute x back into one of the original equations to solve for y. Let's use the first equation: 3x + y = 6 3(1) + y = 6 3 + y = 6Substitute x: Solve for y: 3 + y = 6 y = 6 - 3 y = 3 So, the floor routine is 3 minutes long.

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