may i know the different between linear equation and linear function?

Answer 1

A function f:R^n -> R^m is linear if f(a x + b y) =a f(x)+b f(y) for all real numbers a and b.

A linear equation in the variable x is an equation that can be written in the form
A x = b. (In the general case, A is a matrix; b is a vector--usually assumed to be constant entries. In the general case, x is a vector. Regardless, the entries of A and b do not depend on x.)

Even though the graph of f(x)=3x+2 is a line, f(x)=3x+2 is not a linear function. f(x)=3x is a linear function. 3x+2=0 is a linear equation because we can write it as 3x=-2.

Answer 2

The notions of equation and function are conceptually quite different.

An equation is a mathematical statement expressing that two quantities are equal. For it to be linear, both sides must be a sum of terms that are either just constants or constants times one of the variables to the first power. So the previous examples of y=2x + 6 and 3x + 2 = 6 are both linear equations (the first being a linear equation of 2 variables, the second being a linear equation of 1 variable), while y=sin(x) is not linear because the variable x is inside the complicated sin function, and is not just multiplied by a constant. Other examples of linear equations are 3x+4y=5 and x-y-2z=12 (a linear equation of 3 variables!) The point is that the word "equation" means that the object is an expression that two things are equal, and the word "linear" means that the variables can only occur to the first power. Usually, after setting up an equation, one would either be interested in knowing all possible values of the variables that make the equality hold, or in trying to give a graphical interpretation of these values by making them the coordinates of points. With two variables, pairs (x,y) that satisfy the equation will give you a line in the plane, and with three variables, triples (x,y,z) will give you a plane in 3 dimensional space.

A function is conceptually quite different. You can think of it as a rule whereby one can supply some values for some variables and the rule says to do a particular computation with those supplied values and supply the answer as output. For instance, 3x+2, 2x+6,x^2, and sin(x) are all functions. With 3x+2, if I supply you with a value of x, say 1, then the function is a rule stating that you multiply 1 by 3 and then add two, so the output is 5. To be a linear function, the rule be of the form that you are adding together a sum of terms, each of which is either a constant or a constant times one of the input variables to the first power. So 3x+2, 2x+6 are linear, while x^2 and sin(x) are not.

The difference between a function and an equation can be confusing because if we want to NAME the function, say name it f(x), then we use an equal sign to assign the name to the function, i.e., f(x)=3x+2 gives the function f(x) the name 3x+2. This is a different use of the equal sign than in an equation -- here it is an assignment of a name, and there it is a statement of equality. Because there is this distinction, higher level math texts and computer science programs use the symbol := for assignment, instead of =, so we could write f(x) := 3x+2 to make the distinction clearer.

The other point of confusion between the two concepts is that we often set the value of f(x) to be another variable. For instance, we might set y=f(x)=3x+2, so we get the equation y=3x+2. However, the main reason we do this is because one important interpretation of a function is that it has an associated graph. The graph is found by setting the value of the function equal to a variable, resulting in an equation, and then plotting all points whose coordinates satisfy that equation. Technically, however, y=3x+2 is not the function 3x+2, but the equation associated to the function which gives the graph of the function. Because it does come from a function, however, for an equation that comes from a function like this, x represents the input to the function and y represents the output of the function, so the value of y DEPENDS on the value of x. We say that y is then the dependent variable, while x is the independent variable, because it can be any value that you are allowed to plug into the function. Not all equations need to be thought of as having one of the variables being dependent on the other. 2x+3y=6 by itself is a perfectly valid linear equation, without mention of any linear function.

Summary: Equations are statements of equality, while functions are rules telling you what to do with input numbers. It can get confusing because functions have associated equations, which are usually used to produce the graphs of functions. In both cases, "linear" refers to the expressions being particularly simple, i.e., a sum of terms that are either constants or a constant times a variable to the first power.

Answer 3

1. A linear equation is an equation which is expressed s follows
Y=2X+5
Note that it has two variable, X and Y in the example.
2. A linear function is an equation represented as follows:
f(x)=2X+5:
Y=2X+5
They are the same, but are written differently.

One difference is that when you have a value for X, they are not written the same wy.
Let value of X be 9
Therefore
In an equation, Y=2(9)+5
In a function, f(9)=2(9)+5

Note:
The term linear means that the values of X and Y plotted on a graph would produce a straight line.

Answer 4

Pretty much the same thing... an equation has two sides which have an equivalence.

ie a linear equation is... 3x+2 = 6; thus it has one answer

A function is an equation that is set up as such,

f(x) = 3x+2... where the equation is a function of x, and there can be a multitude of answers.

Also, a linear function does not have to be a straight line, linear only implies that it is a infinitely continuous function unless otherwise specified as piecewise continuous, in which case a function can be linear over only a certain interval.

Answer 5

An equation has an equal sign; a function doesn't.

So y = 3x + 2 is a linear equation, but 3x + 2 is a linear function.

Notice that in the first example, we're setting y equal to the function. That's why you see things like y = f(x) = 3x + 2.

Answer 6

It seems such as you're able to desire to be conscious of the thank you to determine the thank you to get the factors. you're able to desire to make an xy chart. some human beings call it a T chart. you're able to desire to p.c.. 3 numbers for X, then substitute each and every of those into your equation. You multiply via 3 then upload one. in spite of your answer is, that is your y fee. subsequently, I p.c.. the numbers 0, a million, & 2. once you substitute 0 for X, you get y=a million as a results of fact: y = 3x + a million y = 3 (0) + a million y = 0 + a million y = a million once you substitute a million for x, you get y=4 as a results of fact: y = 3x + a million y = 3(a million) + a million y = 3 + a million y = 4 once you substitute 2 for x, you get y=7 as a results of fact: y = 3x + a million y = 3(2) + a million y = 6 + a million y = 7 So your xy chart could look like this: X, Y 0, a million a million, 4 2, 7 So the factors you're able to graph is (0,a million), (a million,4), & (a million,4). hint: it would be a on the instant line. i don't be conscious of the place you acquire (0,4), that factor should not be on the graph.

Answer 7

a linear equation is straight - y=2x+6
a linear function is either straight or curved f(x)=sin x

Answer 8

they are one in the same. the key word really is LINEAR.

this means that superposition must be true. for example:

f(x1) + f(x2) must equal f(x1+x2)
also, A*f(x1) = f(A*x1)

f(x) = x+2 is linear, however f(x) = cos(x) is not