Solve for q.
[tex]$-7 q+12 r=3 q-4 r$[/tex]
Write a formula for [tex]$g(q)$[/tex] in terms of [tex]$q$[/tex].
[tex]$g(q)= \square$[/tex]
Answer (2)
Rearrange the equation − 7 q + 12 r = 3 q − 4 r to isolate terms involving r on one side and terms involving q on the other side.
Combine like terms to simplify the equation to 16 r = 10 q .
Solve for r in terms of q to obtain r = 16 10 q .
Simplify the fraction to get the final answer: r = 8 5 q .
Explanation
Understanding the Problem We are given the equation − 7 q + 12 r = 3 q − 4 r and asked to write a formula for r = g ( q ) in terms of q . This means we need to isolate r on one side of the equation.
Isolating r terms First, let's rearrange the equation to group the terms with r on one side and the terms with q on the other side. We can add 4 r to both sides of the equation: − 7 q + 12 r + 4 r = 3 q − 4 r + 4 r − 7 q + 16 r = 3 q
Isolating q terms Next, we add 7 q to both sides of the equation to isolate the r term: − 7 q + 16 r + 7 q = 3 q + 7 q 16 r = 10 q
Solving for r Now, we can solve for r by dividing both sides of the equation by 16: 16 16 r = 16 10 q r = 16 10 q
Simplifying the fraction Finally, we simplify the fraction 16 10 by dividing both the numerator and the denominator by their greatest common divisor, which is 2: r = 16 ÷ 2 10 ÷ 2 q r = 8 5 q
Final Answer Therefore, the formula for g ( q ) in terms of q is g ( q ) = 8 5 q .
Examples
In physics, if the distance traveled ( r ) is proportional to the time ( q ) with a constant rate, the equation r = g ( q ) = 8 5 q can represent this relationship. For instance, if an object moves at a constant speed such that for every 8 seconds ( q ), it travels 5 meters ( r ), this formula describes that motion. Understanding such relationships is crucial in analyzing motion, calculating speeds, and predicting future positions based on time elapsed. This concept applies to various scenarios, from simple mechanics to more complex systems involving constant rates of change.
The formula for g ( q ) in terms of q is g ( q ) = 8 5 q . To derive this, we rearranged the equation and solved for r . The process involved grouping terms and simplifying the resulting expressions.
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