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Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineGymnasts training for an upcoming competition are practicing their routines for balance beam and floor exercise. During morning practice, Hannah practiced her beam routine 11 time and her floor routine 77 times, which took a total of 1515 minutes. During afternoon practice, she ran through her her beam routine 22 times and her floor routine 22 times, which took a total of 66 minutes. How long is each routine?\newlineThe beam routine is _\_ minutes long and the floor exercise is _\_ minutes long.

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineGymnasts training for an upcoming competition are practicing their routines for balance beam and floor exercise. During morning practice, Hannah practiced her beam routine 11 time and her floor routine 77 times, which took a total of 1515 minutes. During afternoon practice, she ran through her her beam routine 22 times and her floor routine 22 times, which took a total of 66 minutes. How long is each routine?\newlineThe beam routine is _\_ minutes long and the floor exercise is _\_ minutes long.
  1. Define variables: Define the variables for the routines.\newlineLet xx be the time for the beam routine and yy be the time for the floor routine.
  2. Write equations: Write the equations based on the given information.\newlineFor the morning practice, we have:\newline1x+7y=151x + 7y = 15\newlineFor the afternoon practice, we have:\newline2x+2y=62x + 2y = 6
  3. Set up system: Set up the system of equations.\newlineWe have the following system:\newlinex+7y=15x + 7y = 15\newline2x+2y=62x + 2y = 6
  4. Eliminate variable: Decide which variable to eliminate.\newlineWe can eliminate xx by multiplying the first equation by −2-2 and adding it to the second equation.
  5. Perform addition: Multiply the first equation by −2-2 and add it to the second equation to eliminate xx.
    −2(x+7y)=−2(15)-2(x + 7y) = -2(15)
    −2x−14y=−30-2x - 14y = -30
    Now add this to the second equation:
    (2x+2y)+(−2x−14y)=6+(−30)(2x + 2y) + (-2x - 14y) = 6 + (-30)
  6. Simplify equation: Perform the addition to eliminate xx.2x−2x+2y−14y=6−302x - 2x + 2y - 14y = 6 - 300x−12y=−240x - 12y = -24Simplify the equation:−12y=−24-12y = -24
  7. Solve for y: Solve for y.\newlineDivide both sides by −12-12:\newliney=−24−12y = \frac{-24}{-12}\newliney=2y = 2
  8. Substitute and solve: Substitute yy back into one of the original equations to solve for xx. Using the first equation: x+7(2)=15x + 7(2) = 15 x+14=15x + 14 = 15
  9. Final solution: Solve for xx.x+14−14=15−14x + 14 - 14 = 15 - 14x=1x = 1

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