Linear vs. non-linear equations?
352533
Generally speaking, you call "linear" a function F (or an operator whatsoever) which satisfies the following conditions:\n\n1) F(a*X)=a*F(X) for every real number a (X is the argument of the function, whatever it be: it can be a number, a vector, a complex, a function... it doesn't matter)\n\n2) F(X+Y) = F(X)+F(Y) for every X and Y which can be taken as an argument of the function or the operator.\n\nThese two conditions can be sinthesized in the following single one:\n\nCONDITION FOR LINEARITY:\nF(a*X+b*Y) = a*F(X) + b*F(Y) for every a and b reals and for every X and Y in the domain of the function.\n\n----\n\nIn the ring of the polynomials of n variables x1, x2 ... xn, it is easily verified that the only polynomials which can be defined "linear" according to the above conditions are those in which the variables appear at the first degree only, so they are in the form:\n\nP(n) = a1*x1 + a2*x2 + ... + an*xn + a(n+1)\n\nLinear equations in n variables are obtained by setting a linear polynomial of the above type equal to 0.\n\nNon-linear equations are all other possible kinds of equations: the ones obteined by setting to 0 non-linear polynomials (you call them algebraic non-linear equations), or the ones obtained by setting ot 0 expressions which involve logarithms, exponentials etc (in that case you are dealing with transcendant equations)
1016807
**I'm sorry, I really don't know if this is right, but I attempted it. Summer kind of makes you forget stuff...\n\na)If I based which job on pay, the quadratic equation - based pay would be the most logical choice, because of it's characteristis. The basic model, y=x^2+x (**this is what I am not sure on, I don't know if this is the standard equation, I think it is though**), increases, while not steadily, but more rapidly than the basic linear equation, y=x. However depending on other values of the equations, the money could be more on the linear than the quadratic. \n\nb) Yes for instance, if the equation was y=x^2+x, it would be less money than say y=4x, until 2.5 years at the company. Depending on the two equations, one might be worth more than the other, depending on when the two equations meet.\n\nc)No, because both pays are the same and it depends more on the equations than the intitial value.\n\nI gave it my best shot... Took me forever too...Um, 12 minutes...
778614
y =mx + b\n\nFirst, the word linear will tell you this is the answer, as linear graph functions follow this format. From "best expressed", i got the hint that 4) is the answer cos its is the most direct relation. i.e. you plot in x and you get y, like a direct mirror.\n\n1) A quadratic equation will give you 2 solutions to a number you plot in, and from this itself, we can eliminate it. \n2) Complicated. The relationship isnt clear also, cos the entire thing seems to be pure gibberish\n3) This formula can only be applied in a right angled triangle. So it only shows ONE kind of relationship two variables can have.\n\nHope this helps=)
839973
What you have given us is an expression, not an equation.  An equation could be solved for H, however an expression can not.
707230
A linear expression is an expression with ONE variable raised to the order of ONE (that is, the variable is not squared or anything). \n\nIt is important to distinguish between an EXPRESSION and an EQUATION. An equation occurs when something is equal to something (e.g. 1=1, states that 1 is equivalent to 1). \n\nAn expression is simply a string of variables and numbers that represent ONE number (e.g. 1+1, which represents 2). You can tell between the two by the presence of an equal sign (=).\n\nFor example, \n\n1 + x is an EXPRESSION whereas,\n\n1 + x = 2 is an EQUATION. \n\nThe expression 1 + x does not mean a definite number. It is very much like a variable; it can equal just about anything since it simply acts as a placeholder. An equation, on the other hand, explicitly states that the expression is equal to something, thus giving the variable a value.\n\nYou may think of it as a lock and a key. An expression is like a lock, it is simply there and has no meaning until both lock and key are present (equation).\n\nNow, consider your expression 4a + 7 - 2a + 3. The expression, like a lock, is simply there. It has potential, but means nothing. The entire thing could very well be a variable x, and still mean the same thing.\n\nHowever, the expression can be simplified, much a 1 + 3 + 4 can be reduced to 8 (though it will not actually be a number). Gather the like terms (4a - 2a) + (7 +3), then simplify individually:\n2a + 10. This expression is equivalent to the original, and simpler.
752791
you mean f(x)=3x+2, for example?\n\nreplace f(x) with Y, graph and solve as a normal equation. Linear functions have infinite whole solutions, depending on what you want Y to be.\n\nOf course, I could be minunderstanding you completely. If so, sorry.
599765
When graphed, a linear equation will form a straight line while a non linear equation will not be a straight line
789048
It is linear because the current and voltage relationship can be described as a linear differential equation:\n\nI = dQ/dt = C * dV/dt
529628
The degree of polynomial equation is given by the highest power of the variable. Linear equations are of degree one. Quadratic equations are of degree two. As the exponential function is NOT an algebraic funtion (it is trascendental), it cannot be categorized this way.
827005
It's linear alright. x = - 3/7. A little tricky because it's a line parallel to the y axis.
633678
Never Bounded.  Sometimes there is no solution and thus no bounds to the solution set. Sometimes the solution set goes from a number to infinity and thus there is no bounds for the solution set.  Those are the general rules that is.  Although, thinking about it, I think that most linear systems of equations are never totally bounded.  Others are bounded due to curves and shape and things, but I think that linear systems are usually unbounded (when there is only two of them that is).
700548
So x4-5*y2*x2+4*y4=0; or (x^2-y^2)*(x^2-4y^2)=0; or (x-y)*(x+y)*(x-2y)*(x+2y)=0\nIt splits into independent linear equations: (a) y=x; (b) y=-x; (c) y=0.5*x; (d) y=-0.5*x;\nNow exclude (a) & (c); thus (x+y)*(x+2y)=0; did you mean it Don?
895128
If by 'convert' to a linear function you mean approximate by a linear function, then a possible solution is given by the Taylor's expansion of the exponential function.\n\nConsider a point x0, then the linear approximation of y around x0 is given by:\n\ny(x) = e^(x0+a) + e^(x0+a) * (x+a-x0) ,\n\nwhich is linear in x.  Obviously this is only an approximation as the Taylor expansion of the exponential function will involve a very large number of higher order (i.e. quadratic, cubic etc) terms in the right hand side of the equation.
282293
The techniques used to solve systems of linear equations (substitution and elimination) tend to do a pretty good job of solving quadratic systems.\n\nThis page does a pretty good job of demonstrating how:
746522
Equations are used all the time in algebra (linear, quadratic, cubic, exponential, systems of equations and so on).  You can graph equations.  A table can be used to help you graph.  Mathematics is a language itself.\n(I'm not really sure what you are looking for).
700913
a function is written in  the form f(x) or g(X)  or anything relating to that form\nan equation has and equal sign\n\ntherefore this one is a linear equation
1112986
linear: graph is a straight line that is not vertical.  table shows values steadily increasing or decreasing.\n\nquadratic: graph looks like a "U" - either up (v) or down (^) and either wide opening or small opening.  table shows values decreasing/increasing rapidly, then slowly, then the opposite direction, such as:\nx:  1  2 3 4 5 6 7 8  9\ny: 16 8 4 2 1 2 4 8 16\n\nexponential: similar to the quadratic, but it increases more rapidly, has y=1 when x=0, and does not "go the opposite direction" like the quadratic does.  it starts out very little, like a fraction, then shoots up and doesn't come back.  the table is like that: starts out little, increases a little, increases more and more rapidly, then its too big.\n\nabsolute value: looks like a "V", not like a quadratic that looks like a "U".  absolute values have an indifferentiable point, or in other words the bottom tip of the "V".  it's like a linear equation that is positive on one side where it should be negative.  the table is like a linear, but always positive where it should be negative.\n\nsquare root: looks like half a "U" turned 90 degrees clockwise on its side and only positive.  starts along the x-axis with y=0 at some x, then increases rapidly for a short time, then increases very slowly afterwards.\nx: 1  2  3   4    5    6      7     8\ny: 0 .5 .9 1.2 1.4 1.5  1.55 1.57\n\nnote that the sample tables are not representing real functions, they just graph a general idea of what it should look like.