Two machines at a factory are filling cans of soup. One machine has already filled 180 cans of soup, and fills cans at a rate of 15 per minute. The second machine has already filled 210 cans of soup, and fills cans at a rate of 12 per minute. Write an equation to show when the two machines will have filled the same number of cans of soup. Use x as your variable.(1 point)
Responses
12x−210=15x−180
12 x minus 210 equals 15 x minus 180
12x+180=15x+210
12 x plus 180 equals 15 x plus 210
12x+210=15x+180
12 x plus 210 equals 15 x plus 180
210−12x=180−15x
210 minus 12x equals 180 minus 15x
Movie Theatre A has a popcorn machine with a 100-gallon capacity, and sells 3 gallons of popcorn per hour. Movie Theatre B has a popcorn machine with a 150-gallon capacity, and sells 5 gallons of popcorn per hour. Write an equation to show when the two popcorn machines will have the same amount of popcorn left. Use x as your variable.(1 point)
Responses
100+3x=150+5x
100 plus 3 x equals 150 plus 5 x
100−3x=150−5x
100 minus 3 x equals 150 minus 5 x
150+3x=100+5x
150 plus 3 x equals 100 plus 5 x
3x−100=5x−150
3x minus 100 equals 5x minus 150
Linear Equations in Real-World Scenarios Quick Check
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Question
An ice cream shop makes vanilla and chocolate ice cream. The vanilla ice cream machine has a 180-cup capacity, and it sells 20 cups per day. The chocolate ice cream machine has a 140-cup capacity, and it sells 16 cups per day. Write and solve an equation to show when the two machines will have the same amount of ice cream left.(1 point)
Responses
180−20x=140−16x ; x=20
180 minus 20 x equals 140 minus 16 x ; x equals 20
140+16x=180+20x ; x=10
140 plus 16 x equals 180 plus 20 x ; x equals 10
180+16x=140+20x ; x=−10
180 plus 16 x equals 140 plus 20 x ; x equals negative 10
180−20x=140−16x ; x=10
180 minus 20x equals 140 minus 16x ; x = 10
Movie Theatre A has a popcorn machine with a 120-gallon capacity. It sells 7 gallons of popcorn per hour. Movie Theatre B has a popcorn machine with a 150-gallon capacity. It sells 9 gallons of popcorn per hour. Write and solve an equation to show when the two popcorn machines will have the same amount of popcorn left.(1 point)
Responses
120+7x=150+9x ; x=−15
120 plus 7 x equals 150 plus 9 x ; x equals negative 15
120−7x=150−9x ; x=10
120 minus 7 x equals 150 minus 9 x ; x equals 10
120x−7=150x−9 ; x=15
120 x minus 7 equals 150 x minus 9 ; x equals 15
120−7x=150−9x ; x=15
120 minus 7 x equals 150 minus 9 x ; x equals 15
120 minus 7x equals 150 minus 9x ; x = 15
Write an equation and interpret the solution to the following problem: Rental car A has a rate of an initial $100, plus $10 per day. Rental car B has a rate of an initial $50, plus $20 per day. After how many days are the rental car prices equal?(1 point)
Responses
20x−50=10x−100; x=−5; after 5 days the prices are equal
20 x minus 50 equals 10 x minus 100 ; x equals negative 5 ; after 5 days the prices are equal
50+20x=100x+10; x=1; after 1 day the prices are equal
50 plus 20 x equals 100 x plus 10 ; x equals 1 ; after 1 day the prices are equal
50+20x=100+10x; x=5; after 5 days the prices are equal
50 plus 20 x equals 100 plus 10 x ; x equals 5 ; after 5 days the prices are equal
50x+20=100x+10; x=5; after 5 days the prices are equal
50 x plus 20 equals 100 x plus 10 ; x equals 5 ; after 5 days the prices are equal
50 plus 20x equals 100 plus 10x ; x = 5 ; after 5 days the prices are equal
To write an equation to show when the two machines will have filled the same number of cans of soup, we need to set the number of cans filled by each machine equal to each other.
Let x represent the number of minutes elapsed.
The first machine fills cans at a rate of 15 per minute and has already filled 180 cans. Therefore, the number of cans filled by the first machine after x minutes is given by 15x + 180.
The second machine fills cans at a rate of 12 per minute and has already filled 210 cans. Therefore, the number of cans filled by the second machine after x minutes is given by 12x + 210.
Setting these two expressions equal to each other, we have:
15x + 180 = 12x + 210
This equation represents the point at which both machines will have filled the same number of cans of soup.