How to linearize equations to y = mx + b form (more information below)?

[V21]

This question is mainly asking to rewrite a different graph equation (maybe parabolic, curved, etc) and changing it to a normal linear equation in the form y = mx+b. The question 5 and 6 can be seen in the picture, with further questions asking:

i) Determine the dependent and independent variable. ii) Re-arrange the equation so that it is in the form of the straight-line equation. iii) What is the gradient and y-intercept of this re-arranged equation? iv) Sketch what the straight-line graph would look like.

I know that linearisation works by first finding out what the y and x variables and then leaving x alone along with m multiplied to it. This works for simple examples like F = ma, where m is the gradient, a is x and F is y. When plotting on a graph, it will be F on the y axis and a on the x axis. However, for the fifth and sixth question, I'm not so sure how to do it. My first question is how to find out what is the independent and dependent variable in an equation with many variables and then how to re-arrange it to y = mx+b form. I thought for the fifth question the independent variable would be the length and the dependent variable the frequency. Then, frequency can be represented by 1/f. This would make f = 4/v * (l+e). This would give the gradient as 4/v I think and the x axis will be length, y axis will be f. I am not so sure. For the sixth question, I assumed v to be the dependent variable and u to be the independent. I made v = fu/u-f but I don't know whether any of these steps are correct and if they are, how to change it to the y = mx+ b format

[studiot]

I am assuming you want to achieve a straight line of the form y = mx + b, where m and b are constants.

Strange though it may seem, this is not actually a 'linear' equation, so it is better not to use that description.
'Linear' has a special meaning in maths.

y = mx is linear

y= mx + b is what is known as affine.

 

Back to your questions.

starting with number 5 

Your two variables are length L and frequency. f.

the velocity, v and end correction, e are stated to be constants.

But there is an inverse relationship between variables L and f.

So you need to somehow introduce a new variable by inverting one of these.

Can you think of a relationship (another equation) for either L or f that will do this?

[V21]

Thank you for your reply. What do you mean by inverting one of the variables? Does it mean that L = 1/f? Also, how do you find out this is the way to solve such kind of sums? 

[studiot]

1 hour ago, V21 said:

Thank you for your reply. What do you mean by inverting one of the variables? Does it mean that L = 1/f? Also, how do you find out this is the way to solve such kind of sums? 

If one variable decreases as the other increases then they have an inverse relationship.

Is this not true of L and f thus?

[math]{\rm{L(plus}}\,{\rm{a}}\,{\rm{constant)}}\quad {\rm{ = }}\quad \frac{{{\rm{(a}}\,{\rm{constant)}}}}{{\rm{f}}}[/math]

 

So do you know any relationship (for sound waves ?) involving any of the quantities in the original given equation ?

You are looking for an equation of the form

[math]{\rm{something = }}\frac{{{\rm{something}}\,{\rm{else}}}}{{\rm{f}}}[/math]

 

 

Edited August 23, 2019 by studiot