I don't understand how to full explain how to get a solution.

We recently covered 'enough to make' constraints in my further maths class. Although I do know how to do the question (in a sense) I am unable to explain how in an mathematical terms, which does not include skipping steps (which I don't know) to instead do ones which I know will be included and accurate. This is an issue as even my teacher doesn't know exactly what is going on, I spent 30 mins after school explaining my thought process for him to still be confused and having a vey shaky rough way of making the question work.
The example we was given was:
"Karen has enough eggs to make 10 chocolate cakes or 6 strawberry cakes"
We are to let chocolate cakes t be x, and strawberry cakes to be y.
We need to write this as a linear inequality. When we do this we get 3x + 5y <(or equal to) 30, which is correct.
I do not know how to explain the process of working behind this and if anyone could explain it mathematically that would be very helpful, thankyou.

We recently covered 'enough to make' constraints in my further maths class. Although I do know how to do the question (in a sense) I am unable to explain how in an mathematical terms, which does not include skipping steps (which I don't know) to instead do ones which I know will be included and accurate. This is an issue as even my teacher doesn't know exactly what is going on, I spent 30 mins after school explaining my thought process for him to still be confused and having a vey shaky rough way of making the question work.
The example we was given was:
"Karen has enough eggs to make 10 chocolate cakes or 6 strawberry cakes"
We are to let chocolate cakes t be x, and strawberry cakes to be y.
We need to write this as a linear inequality. When we do this we get 3x + 5y <(or equal to) 30, which is correct.
I do not know how to explain the process of working behind this and if anyone could explain it mathematically that would be very helpful, thankyou.

Are you sure it's 3x + 5y <(or equal to) 30 ?
I'm getting 3x + 5y <= 60 .... I'll try again!

We recently covered 'enough to make' constraints in my further maths class. Although I do know how to do the question (in a sense) I am unable to explain how in an mathematical terms, which does not include skipping steps (which I don't know) to instead do ones which I know will be included and accurate. This is an issue as even my teacher doesn't know exactly what is going on, I spent 30 mins after school explaining my thought process for him to still be confused and having a vey shaky rough way of making the question work.
The example we was given was:
"Karen has enough eggs to make 10 chocolate cakes or 6 strawberry cakes"
We are to let chocolate cakes t be x, and strawberry cakes to be y.
We need to write this as a linear inequality. When we do this we get 3x + 5y <(or equal to) 30, which is correct.
I do not know how to explain the process of working behind this and if anyone could explain it mathematically that would be very helpful, thankyou.

MQB is perfectly right I believe, it's a bit of a funny question because you need to solve it as you would a linear graph and you might think of it just as an inequality. The idea of the eggs is a bit of a red herring. According to the question, when we have 10 chocolate cakes (x) we have 0 strawberry cakes. When we have 6 strawberry cakes (y) we have 0 chocolate cakes. So, just for easiness (and as the former is x) place chocolate cakes along the x axis, and plot these points out. You'll note that when x=0, y=6, so therefore 6 is the y-intercept - think in terms of y=mx+c. So, to get me, we use the formula
$\frac{y_2-y_1}{x_2-x_1}$
Therefore:
$\frac{0-6}{10-0} = -0.6$
So our completed equation for the line is:
$y=-0.6x+6$
So you'll note that if we input the x and y values we used before, we'd get the number of both cakes possible at that point. So, while it might be odd to have done this as a graph, what we've actually done is just found an equation that links x and y, by visualizing it as a graph. What does this have to do an inequality? So You'll also note that because we don't have many eggs, the domain of the graph is
$0 \leq x \leq 10$
and the range (y values) are:
$0 \leq f(x) \leq 6$
Remember f(x) is just the output values, so y. So the highest value of x is 10, for example. If we input this into the equation, y is 0 at this point. So then again, as we're looking at how many cakes we can bake, we don't always have to bake 10 chocolate cakes, for example: we might want 9, or less. However, we cannot go above 10 because we don't have enough eggs. If we rephrase that for the equation:
$y \leq -0.6x+6$
And then we rearrange:
$[br]y+0.6x \leq 6[br]5y+3x \leq 30[br]$
We always need to represent inequalities in their smallest form, but the coefficient of x must be an integer. The lowest integer it can be is 3, when we multiply it by 5, so we multiply the entire equation and get that answer.
I realise this is just rephrasing the former users' answer, and very much kudos to them, but I find it often helps to see multiple explanations sometimes Good luck!

Pretty much agree though for the OP its probably worth noting that you can interpret the
3x+5y<=30
in terms of the number of eggs, at least in a simple case. It takes 3 eggs to make a chocolate cake and 5 eggs to make a strawberry, then 30 represents the total number of eggs available. If you made 9 chocolate cakes instead of 10, then youd have 3 eggs spare so not enough to make 1 strawberry cake which would requre 27+5=32 eggs which voilates the constraint. The 3/5 gradient represents the cake egg ratio and the 6*5 represents the total number of eggs. Thinking about 0.6 = 3/5 of a cake is ok mathematically but its worth noting that really its an integer constraint / programming problem.

This basic idea still holds in the more general case.