Math  /  Algebra

QuestionWrite a system of equations to describe the situation below, solve using any method, and fill in the blanks.
Oscar and Roger started out at their houses and are biking towards each other. Oscar started out first, and has already gone 4 kilometers. He bikes at a constant speed of 4 kilometers per hour. Roger just left, and rides at 6 kilometers per hour. When the boys meet halfway between their houses, they will continue to the park together. How far will each boy have ridden? How long will that take?
Oscar and Roger will have each biked \square kilometers in \square hours.

Studdy Solution

STEP 1

1. Oscar and Roger are biking towards each other from their respective houses.
2. Oscar has already traveled 4 kilometers before Roger starts.
3. Oscar's biking speed is 4 kilometers per hour.
4. Roger's biking speed is 6 kilometers per hour.
5. They meet halfway between their houses.

STEP 2

1. Define variables and set up equations.
2. Write the system of equations.
3. Solve the system of equations.
4. Calculate the distance each boy has ridden and the time taken.

STEP 3

Define variables: Let t t be the time in hours after Roger starts biking until they meet. Let d d be the distance from Oscar's starting point to the meeting point.

STEP 4

Write the system of equations based on the given information:
1. Oscar's distance equation: d=4+4t d = 4 + 4t
2. Roger's distance equation: d=6t d = 6t

Since they meet at the same point, the distances are equal: 4+4t=6t 4 + 4t = 6t

STEP 5

Solve the system of equations:
Subtract 4t 4t from both sides of the equation: 4=6t4t 4 = 6t - 4t 4=2t 4 = 2t
Divide both sides by 2 to solve for t t : t=42 t = \frac{4}{2} t=2 t = 2

STEP 6

Calculate the distance each boy has ridden and the time taken:
Substitute t=2 t = 2 back into one of the distance equations to find d d : Using Roger's equation: d=6t d = 6t d=6×2 d = 6 \times 2 d=12 d = 12
So, each boy will have biked 12 kilometers.
The time taken is t=2 t = 2 hours.
Oscar and Roger will have each biked 12 \boxed{12} kilometers in 2 \boxed{2} hours.

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