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What the difference between Non linear problem and linear problem in simple english.

I'd like to get an example in words and explanation why it is a linear/non linear problem.

I want the example to be something like 

"You have to paint a full house when every room consumes x amount of paint and you can only buy paint in

3 gallons for $15

7 for $30

20 for $85

how many of each of the options should i buy in order we should have enough to paint the full house and it should cost the least.

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5 Answers

If x and y have a linear relation, (sometime also read as "x is directly proportional to y,") then multiplying x by some (non-zero) amount we result in multiplying y by the same amount.

E.g. 8 ounces = 1 cup, and 32 ounces = 4 cups

E.g. The cost you pay for filling a tank of gas is directly proportional to the number of gallons put in the tank


(Note: NOT all linear problems have the property that when x = 0, y = 0 too, (but my two examples do.))

 

Non-linear problems do not have the above property.

E.g. The time an object has been falling, and the distance it has fallen are NOT directly directly proportional.

When the maximum power of all variables in a term (added together) in an equation is 1 it is linear. When you have an equation with maximum power (added together in a term)  of the variables not equal to 1,  it is non-linear.

Example: y = mx + b, ax + by + c =0 are linear equatiions

ax2 + bx + c  = y  or xy + 3x + 4=0 are non-linear. ( in the second equation the term xy has a power of two when powers of x and y are added)

A linear problem is any problem which is solved by setting up only linear equations or linear systems of equations to solve. An expression in the variables x1,...,xn is linear if it is of the form a1x1+...+anxn+b. Here a1,...an and b are coefficients.

If the expressions on both sides of the equation are of this form then it is a linear equation.

If a problem requires nonlinear equations to solve then it is a nonlinear problem.

For example xy=1 is not linear, as is e^x+sin(y)=ln(x+y) and may only arrise in solving nonlinear problems.

It is a linear problem if you can plot out the answers that fit x and y and draw a straight line through these coordinates. I think that's as simple as you can get for an answer. 

A linear problem is a line and will not contain x or y to any power greater than or less than 1.
y=x is a good example of a linear problem (because the solution is a line)
y=x^2 is a non-linear problem