Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ In a paint factory, an old conveyer line has filled $17$ barrels of paint, and is filling more at a rate of $8$ barrels per minute. A worker just switched on a newer line that can fill $9$ 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? $\newline$ In $\_\_\_$ minutes, both lines will have filled $\_\_\_$ barrels.

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Solve a system of equations using any method: word problems

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

\newline

In a paint factory, an old conveyer line has filled

17

barrels of paint, and is filling more at a rate of

8

barrels per minute. A worker just switched on a newer line that can fill

9

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?

\newline

\_\_\_

minutes, both lines will have filled

\_\_\_

barrels.

Define variables: Let's define the variables: Let $x$ 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 $17$ barrels and continues to fill at a rate of $8$ barrels per minute. The new line fills at a rate of $9$ barrels per minute.
Write old line equation: Write the equation for the old line: The old line has a head start of $17$ barrels and fills $8$ barrels per minute. So, the total number of barrels filled by the old line after $x$ minutes is $17 + 8x$ .
Write new line equation: Write the equation for the new line: The new line starts from $0$ barrels and fills $9$ barrels per minute. So, the total number of barrels filled by the new line after $x$ minutes is $9x$ .
Set up equal barrels equation: Set up the equation where both lines have filled an equal number of barrels: $17 + 8x = 9x$ .
Solve for x: Solve for x: Subtract $8x$ from both sides to find the value of $x$ . $\newline$ $17 + 8x - 8x = 9x - 8x$ $\newline$ $17 = x$
Substitute $x$ into new line equation: Now that we have the value of $x$ , we can find out how many barrels each line has filled. Since $x$ represents the number of minutes, we can substitute $x$ into either equation. Let's use the new line's equation: $9x = 9(17)$ .
Calculate barrels filled by new line: Calculate the total number of barrels filled by the new line: $9(17) = 153$ barrels.
Calculate barrels filled by old line: Since both lines have filled an equal number of barrels, the old line has also filled $153$ barrels.