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

Question

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.

Answer 1

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. 

Answer 2

In a linear problem there is a constant of variation that is consistent with each data point. For example “If you earn $15 per hour, you earn $30 in 2 hours. Therefore the constant of variation is $15. Whereas in a non linear problem, the constant of variation changes from on data point to another. An example of a non-linear problem is y=x^2. If you begin with x=1,2,3,4 the resulting y=1,4,9,16. The first constant of variation changes from 3 to 5 to 7 as x increases. By using a graphing calculator or a graphing utility, if you graph y=x the result is a line, but if you graph y=x^2 the result is a curve. Another example of a non-linear problem is y=2^x. Graph this one and see how these non-linear problems differ.

Answer 3

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)

Answer 4

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,...,x_n is linear if it is of the form a₁x₁+...+a_nx_n+b. Here a₁,...a_n 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.

Answer 5

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.

Answer 6

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