y=3x+4 is this equation linear or nonlinear? Using 2 properties of linearity?
If you mean the mapping y(x) := 3x+4, then y(x) is not linear, because it does not satisfy the two properties of linearity. Only equations of the form y(x) := ax (for "a" being any real or complex number) are linear.
Don't be confused with the plot of the graph of y(x) = 3x+4 being a line - this does not mean linear in the sense of the definition in Linear Algebra. Some authors define such function as "affine-linear".
If you mean the mapping y(x) := 3x+4, then y(x) is not linear, because it does not satisfy the two properties of linearity. Only equations of the form y(x) := ax (for "a" being any real or complex number) are linear.
Don't be confused with the plot of the graph of y(x) = 3x+4 being a line - this does not mean linear in the sense of the definition in Linear Algebra. Some authors define such function as "affine-linear".
y=3x+4 is a linear equation as all components are linear in terms of x (i.e. the only exponents of x are 0 or 1).
However, if you mean a linear map satisfying the additivity and homogeneity properties, y(x) nonlinear as described by Armin above.
It is a linear equation because the maximum exponent of X is one in this equation. This is the same as famous line equation y = mx+c, in which 3 is the slope of line and 4 is the y intercept.
If we are talking bout Linear Algebra, the equation y(x)=3x+4 is linear and the graphic is a straight line in the plane.
The general structure of a linear equation is y(x)=mx+b
Actually you have an affine function or affine transformation, see links below.
http://mathworld.wolfram.com/AffineFunction.html
https://en.wikipedia.org/wiki/Affine_transformation
A linear function is in the form y = mx + b or f(x) = mx + b, where m is the slope or rate of change and b is the y-intercept or where the graph of the line crosses the y axis. You will notice that this function is degree 1 meaning that the x variable has an exponent of 1. If a function is nonlinear, then the exponent of the x variable would have an exponent of something other than 0 or 1. Another linear function is in the form of y = a or f(x) = a, where is any real number. The exponent of x in this function is 0 because x^0 = 1, ie y = ax^0 or y = 1x.
This is a linear function. Not all linear function cross the origen, as this case, because b=4. the important is that the exponent in this case of variable x is 1, if is greather than 1 will be not linear.
As pointed out by Peter above, a linear function is a polynomial of degree one. Therefore, ax +by = c or ax + by + cz = d are linear functions. If you talk about linear transformation or linear operator then the story is different.
Definition. A linear equation is an equation of the form
a1x1+a2x2+…+anxn = b,
where x1,x2,…,xn are variables (unknowns) and a1,a2,…,an, b are constants.
Therefore
y=3x+4 if and only if y - 3x = 4,
by definition it is a linear equation.
It is clear that equation y=3x+4 is a linear equation:
A common form of a linear equation in the two variables x and y is y=mx + b,
where m and b designate constants (parameters). The origin of the name "linear" comes from the fact that the set of solutions of such an equation forms a straight line in the plane. In this particular equation, the constant m (in this case 3) determines the slope or gradient of that line, and the constant term b (in this case 4) determines the point at which the line crosses the y-axis, otherwise known as the y-intercept.
Since terms of linear equations cannot contain products of distinct or equal variables, nor any power (other than 1) or other function of a variable, equations involving terms such as xy, x^2, y^(1/3), and sin(x) are nonlinear.
Dear safeer Ullah,
A lot has been spoken about the question in hand. I may not be able to add something more informative, especially considering the comments of Prof. Armin Fügenschuh and Mr. Jose William Porras on each dimension.
Well my take is: by definition it is not linear, a simple "no" against your question. But practically speaking, y is a linear function of x. A good explanation can be found in the context of affine transformation as correctly suggested by Ms. Hanifa Zekraoui. Unlike the linear transformation, it takes into account the transformation of the fixed origin. Therefore, if the origin is transformed to (0,0) then the system will look like what it is.
Moreover, y=3x+4 is a non homogeneous version of y=3x. For example in linear control systems, y=Cx+D is called as a linear output.
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