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Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineTo keep in shape, Julie exercises at a track near her home. She requires 6262 minutes to do 44 laps running and 1010 laps walking. In contrast, she requires 6565 minutes to do 55 laps running and 1010 laps walking. Assuming she maintains a consistent pace while running and while walking, how long does Julie take to complete a lap?\newlineJulie takes _____ minutes to run a lap and _____ minutes to walk a lap.

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineTo keep in shape, Julie exercises at a track near her home. She requires 6262 minutes to do 44 laps running and 1010 laps walking. In contrast, she requires 6565 minutes to do 55 laps running and 1010 laps walking. Assuming she maintains a consistent pace while running and while walking, how long does Julie take to complete a lap?\newlineJulie takes _____ minutes to run a lap and _____ minutes to walk a lap.
  1. Define variables: Let's define two variables: let rr be the time in minutes it takes Julie to run a lap, and ww be the time in minutes it takes Julie to walk a lap. We can then write two equations based on the information given:\newlineFor the first scenario (44 laps running and 1010 laps walking taking 6262 minutes):\newline4r+10w=624r + 10w = 62\newlineFor the second scenario (55 laps running and 1010 laps walking taking 6565 minutes):\newline5r+10w=655r + 10w = 65
  2. Write equations: Now we will use the elimination method to solve the system of equations. To eliminate one of the variables, we can multiply the first equation by −5-5 and the second equation by 44, so that when we add the two equations, the terms with rr will cancel out.\newlineMultiplying the first equation by −5-5:\newline−5(4r+10w)=−5(62)-5(4r + 10w) = -5(62)\newline−20r−50w=−310-20r - 50w = -310\newlineMultiplying the second equation by 44:\newline4(5r+10w)=4(65)4(5r + 10w) = 4(65)\newline20r+40w=26020r + 40w = 260
  3. Elimination method: Next, we add the two resulting equations to eliminate rr:\newline(−20r−50w)+(20r+40w)=−310+260(-20r - 50w) + (20r + 40w) = -310 + 260\newline−20r+20r−50w+40w=−310+260-20r + 20r - 50w + 40w = -310 + 260\newline0r−10w=−500r - 10w = -50\newlineNow we can solve for ww by dividing both sides by −10-10:\newline−10w−10=−50−10\frac{-10w}{-10} = \frac{-50}{-10}\newlinew=5w = 5\newlineJulie takes 55 minutes to walk a lap.
  4. Add equations: Now that we have the value for ww, we can substitute it back into one of the original equations to find rr. We'll use the first equation:\newline4r+10w=624r + 10w = 62\newline4r+10(5)=624r + 10(5) = 62\newline4r+50=624r + 50 = 62\newlineSubtract 5050 from both sides to solve for rr:\newline4r=62−504r = 62 - 50\newline4r=124r = 12\newlineDivide both sides by 44:\newlinerr00\newlinerr11\newlineJulie takes rr22 minutes to run a lap.

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