Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ In a paint factory, an old conveyer line has filled $10$ barrels of paint, and is filling more at a rate of $1$ barrel per minute. A worker just switched on a newer line that can fill $6$ barrels per minute. In a little while, the two lines will have filled an equal number of barrels. How many barrels will each line have filled? How long will that take? $\newline$ Both lines will have filled $\_$ barrels in $\_$ minutes.

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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

In a paint factory, an old conveyer line has filled

10

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

1

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

6

barrels per minute. In a little while, the two lines will have filled an equal number of barrels. How many barrels will each line have filled? How long will that take?

\newline

Both lines will have filled

\_

barrels in

\_

minutes.

Define Variables: Let's define the variables: Let $x$ be the number of minutes after the newer line starts. The old line has already filled $10$ barrels and continues to fill at a rate of $1$ barrel per minute. The newer line fills at a rate of $6$ barrels per minute.
Write Equations: Write the equations for each line: $\newline$ For the old line: $y = 10 + x$ (since it starts with $10$ barrels and fills $1$ more each minute). $\newline$ For the new line: $y = 6x$ (since it starts from $0$ and fills $6$ barrels per minute).
Set Equations Equal: Set the equations equal to solve for $x$ : $\newline$ $10 + x = 6x$ . $\newline$ Rearrange to solve for $x$ : $\newline$ $10 = 5x$ . $\newline$ $x = 2$ minutes.
Substitute and Solve: Substitute $x = 2$ back into either equation to find $y$ : $\newline$ Using $y = 6x$ : $\newline$ $y = 6(2) = 12$ barrels.