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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, Danny exercises at a track near his home. He requires 3434 minutes to do 99 laps running and 44 laps walking. In contrast, he requires 4040 minutes to do 1010 laps running and 55 laps walking. Assuming he maintains a consistent pace while running and while walking, how long does Danny take to complete a lap?\newlineDanny 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, Danny exercises at a track near his home. He requires 3434 minutes to do 99 laps running and 44 laps walking. In contrast, he requires 4040 minutes to do 1010 laps running and 55 laps walking. Assuming he maintains a consistent pace while running and while walking, how long does Danny take to complete a lap?\newlineDanny takes _____ minutes to run a lap and _____ minutes to walk a lap.
  1. Define Equations: Let's denote the time Danny takes to run a lap as rr minutes and the time he takes to walk a lap as ww minutes. We can then write two equations based on the information given:\newlineFor 99 laps running and 44 laps walking taking 3434 minutes: 9r+4w=349r + 4w = 34\newlineFor 1010 laps running and 55 laps walking taking 4040 minutes: 10r+5w=4010r + 5w = 40
  2. Use Elimination Method: To use elimination, we want the coefficients of one of the variables to be the same (or opposites) in both equations. We can multiply the first equation by 55 and the second equation by 44 to get the coefficients of ww to match:\newline(9r+4w)×5=34×5(9r + 4w) \times 5 = 34 \times 5\newline(10r+5w)×4=40×4(10r + 5w) \times 4 = 40 \times 4
  3. Multiply Equations: After multiplying, we get the new system of equations:\newline45r+20w=17045r + 20w = 170\newline40r+20w=16040r + 20w = 160
  4. Eliminate Variable: Now we can subtract the second equation from the first to eliminate ww:(45r+20w)−(40r+20w)=170−160(45r + 20w) - (40r + 20w) = 170 - 160
  5. Solve for r: This simplifies to: 5r=105r = 10
  6. Substitute rr: Dividing both sides by 55 to solve for rr gives us:\newliner=105r = \frac{10}{5}\newliner=2r = 2\newlineSo, Danny takes 22 minutes to run a lap.
  7. Solve for w: Now we can substitute r=2r = 2 into one of the original equations to solve for ww. We'll use the first equation:\newline9r+4w=349r + 4w = 34\newline9(2)+4w=349(2) + 4w = 34
  8. Solve for w: Now we can substitute r=2r = 2 into one of the original equations to solve for ww. We'll use the first equation:\newline9r+4w=349r + 4w = 34\newline9(2)+4w=349(2) + 4w = 34 This simplifies to:\newline18+4w=3418 + 4w = 34
  9. Solve for w: Now we can substitute r=2r = 2 into one of the original equations to solve for ww. We'll use the first equation:\newline9r+4w=349r + 4w = 34\newline9(2)+4w=349(2) + 4w = 34 This simplifies to:\newline18+4w=3418 + 4w = 34 Subtracting 1818 from both sides gives us:\newline4w=34−184w = 34 - 18\newline4w=164w = 16
  10. Solve for w: Now we can substitute r=2r = 2 into one of the original equations to solve for ww. We'll use the first equation:\newline9r+4w=349r + 4w = 34\newline9(2)+4w=349(2) + 4w = 34 This simplifies to:\newline18+4w=3418 + 4w = 34 Subtracting 1818 from both sides gives us:\newline4w=34−184w = 34 - 18\newline4w=164w = 16 Dividing both sides by 44 to solve for ww gives us:\newlineww00\newlineww11\newlineSo, Danny takes 44 minutes to walk a lap.

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