Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Felipe and Philip started out at their houses and are biking towards each other. Felipe started out first, and has already gone $5$ kilometers. He bikes at a constant speed of $6$ kilometers per hour. Philip just left, and rides at $7$ 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? $\newline$ Felipe and Philip will have each biked $\_$ kilometers in $\_$ hours.

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Solve a system of equations using substitution: word problems

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Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.

\newline

Felipe and Philip started out at their houses and are biking towards each other. Felipe started out first, and has already gone

5

kilometers. He bikes at a constant speed of

6

kilometers per hour. Philip just left, and rides at

7

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?

\newline

Felipe and Philip will have each biked

\_

kilometers in

\_

hours.

Define variables and equations: Let's define the variables and write the system of equations. Let $x$ be the time in hours that Felipe rides until they meet, and $y$ be the time in hours that Philip rides until they meet. Since Felipe has already gone $5$ kilometers, we can write Felipe's distance as $5 + 6x$ (because he rides at $6$ kilometers per hour). Philip's distance is $7y$ (because he rides at $7$ kilometers per hour). They meet halfway between their houses, so the distances they ride must be equal. Therefore, we can write the equation: $\newline$ $5 + 6x = 7y$
Second equation for system: Now we need a second equation to solve the system. Since they meet halfway and continue to the park together, the time they spend biking until they meet must be the same. Therefore, we can write the equation: $\newline$ $x = y$
Substitution to solve system: We can now use substitution to solve the system. Since $x = y$ , we can substitute $y$ for $x$ in the first equation: $\newline$ $5 + 6x = 7x$
Solve for x: Now we solve for x: $\newline$ $5 = 7x - 6x$ $\newline$ $5 = x$
Find distance for Felipe: Since $x = y$ , we also have: $\newline$ $y = 5$
Find distance for Philip: Now we can find the distance each boy has ridden. For Felipe: $\newline$ Distance = $5 + 6x$ $\newline$ Distance = $5 + 6(5)$ $\newline$ Distance = $5 + 30$ $\newline$ Distance = $35$ kilometers
Final answer: For Philip: $\newline$ Distance = $7y$ $\newline$ Distance = $7(5)$ $\newline$ Distance = $35$ kilometers
Final answer: For Philip: $\newline$ Distance = $7y$ $\newline$ Distance = $7(5)$ $\newline$ Distance = $35$ kilometersNow we can fill in the blanks with the final answer: $\newline$ Felipe and Philip will have each biked $35$ kilometers in $5$ hours.