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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineIn a paint factory, an old conveyer line has filled 1717 barrels of paint, and is filling more at a rate of 88 barrels per minute. A worker just switched on a newer line that can fill 99 barrels per minute. In a little while, the two lines will have filled an equal number of barrels. How long will that take? How many barrels will each line have filled?\newlineIn ___\_\_\_ minutes, both lines will have filled ___\_\_\_ barrels.

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Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineIn a paint factory, an old conveyer line has filled 1717 barrels of paint, and is filling more at a rate of 88 barrels per minute. A worker just switched on a newer line that can fill 99 barrels per minute. In a little while, the two lines will have filled an equal number of barrels. How long will that take? How many barrels will each line have filled?\newlineIn ___\_\_\_ minutes, both lines will have filled ___\_\_\_ barrels.
  1. Define variables: Let's define the variables: Let xx be the number of minutes after the new line starts that the two lines will have filled an equal number of barrels. The old line has already filled 1717 barrels and continues to fill at a rate of 88 barrels per minute. The new line fills at a rate of 99 barrels per minute.
  2. Write old line equation: Write the equation for the old line: The old line has a head start of 1717 barrels and fills 88 barrels per minute. So, the total number of barrels filled by the old line after xx minutes is 17+8x17 + 8x.
  3. Write new line equation: Write the equation for the new line: The new line starts from 00 barrels and fills 99 barrels per minute. So, the total number of barrels filled by the new line after xx minutes is 9x9x.
  4. Set up equal barrels equation: Set up the equation where both lines have filled an equal number of barrels: 17+8x=9x17 + 8x = 9x.
  5. Solve for x: Solve for x: Subtract 8x8x from both sides to find the value of xx.\newline17+8x−8x=9x−8x17 + 8x - 8x = 9x - 8x\newline17=x17 = x
  6. Substitute xx into new line equation: Now that we have the value of xx, we can find out how many barrels each line has filled. Since xx represents the number of minutes, we can substitute xx into either equation. Let's use the new line's equation: 9x=9(17)9x = 9(17).
  7. Calculate barrels filled by new line: Calculate the total number of barrels filled by the new line: 9(17)=1539(17) = 153 barrels.
  8. Calculate barrels filled by old line: Since both lines have filled an equal number of barrels, the old line has also filled 153153 barrels.

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