Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. Band students are tested on, and required to pass, a certain number of... Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. Band students are tested on, and required to pass, a certain number of scales during the year. As of today, Kenji has passed 8 scales, whereas his friend Ava has passed 11 of them. Going forward, Kenji has committed to passing 3 scales per week, and Ava has committed to passing 2 per week. At some point soon, the two friends will have passed the same number of scales. How long will that take? How many scales will that be?
Understand the Problem
The question is asking to formulate a system of equations based on the scenario provided about two friends, Kenji and Ava, who are passing scales at different rates. We need to define the equations that represent their progress, solve for the time it takes for them to pass the same number of scales, and find out how many scales that will be.
Answer
In $3$ weeks, the friends will have passed $17$ scales.
Answer for screen readers
In ( 3 ) weeks, the friends will have passed ( 17 ) scales.
Steps to Solve
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Define Variables Let ( t ) be the number of weeks that will pass. We need to express the total number of scales passed by each friend after ( t ) weeks.
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Create Equations for Each Friend
- Kenji's total after ( t ) weeks: $$ K(t) = 8 + 3t $$
- Ava's total after ( t ) weeks: $$ A(t) = 11 + 2t $$
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Set Up the Equation for Equality To find out when they will have passed the same number of scales: $$ K(t) = A(t) $$
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Substitute and Solve for ( t ) Set the equations equal to each other: $$ 8 + 3t = 11 + 2t $$
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Isolate ( t ) Subtract ( 2t ) from both sides: $$ 8 + t = 11 $$
Subtract 8 from both sides: $$ t = 3 $$
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Calculate Total Scales Passed Now, substitute ( t = 3 ) into either equation: For Kenji: $$ K(3) = 8 + 3(3) = 17 $$
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Fill in the Blanks In ( 3 ) weeks, the friends will have passed ( 17 ) scales.
In ( 3 ) weeks, the friends will have passed ( 17 ) scales.
More Information
Kenji and Ava's progress in passing scales is a great way to understand how linear equations model real-life situations. By setting up equations based on their rates, we can predict future outcomes.
Tips
- Forgetting to add the initial number of scales passed when creating the equations.
- Incorrectly simplifying equations; always double-check your math.
- Mixing up the rates at which Kenji and Ava pass scales.
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