What is the difference between an equation, a linear equation and a function?
818092
When Graphed\n\nEquations can be any line\nLinear Equations form only straight lines\nFunction is only a function if you don't have 2 identical x values.
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.
763572
Hi!\n\nSo, what you are looking at is a linear equation.  You can tell this because there is only one instance of the variable x, and it is not raised to any power.  So, the graph will take the form of a line.\n\nAll linear equations take the form f(x) = mx + b\n\nWhere m is typically called the "slope" and b is called the "y-intercept".   The y-intercept is the point on the graph where the function intercepts the y-axis (the one that goes from top to bottom on the graph).  So, right away you can tell that the y-intercept in your equation is -5 (negative 5).  What this means is that the graph intercepts the y-axis at the -5, or in other words, there is a point on the graph at the location (0,-5).  You can put a dot there.\n\nThe next thing you need to see is the slope.  In you equation, the slope is -2 (negative 2).   Slope really means "rise over the run", or in other words, how much the graph rises for every point it goes to the right.  A slope of negative 2 means that for every increase in x of one unit, the y value decreases by negative 2.   \n\nSo, starting at the point (0,-5) which we determined above, go down 2 units and to the right one unit (putting you at the point (1,-7).  That's another point.  You can do it again and find the next point which is (2,-9) and put another point.  You now can put your ruler on the graph and draw the rest of it.\n\nIt's good to understand how this type of equation works.  You could have also made a table of x values and y values, and simply computed a series of y values for a series of x values (that you picked).  Then you could have drawn the graph also.  But understanding this type of equation will help you in the future.
818907
Linear means that the relationship between two variables varies in the same way, except by a constant factor (A) and a constant offset (B).  y = A*x + B\n\nThe volume of water in a tank, and its mass are linearly related.  [m = p*V, where m is the mass, p (rho) is the density of water, which is constant (converted to the units of m and V), and V is the volume of water.  The units of m and V are usually kg and liters, or lbs and ft^3.]\n\nAny other kind of relationship is non-linear, which includes most every process in the real world.  Non-linear relationships usually seem to result from friction/entropy/inefficiency or from forces like magnetic fields, electric fields, or gravity, or well pretty much everything real.
138490
Infinite.\nIn this case all variables and constants have canceled out, leaving a trivially true statement. The original equation, therefore, would be called an identity and one would not consider the graph (it would be the entire xy-plane). An example is 2x + 4y = 2(x + 2y). The two expressions on either side of the equal sign are always equal, no matter what values are used for x and y.
444364
They are the values that make the equation true.  If the equation is true then the equation represents a straight line.
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First equation:\n(x+a+b)^2 = x^2 + (x+a)^2 - (x+b)^2\nor x^2 + 2x(a+b) + (a+b)^2 = x^2 + x^2 + 2ax + a^2 - x^2 - 2bx - b^2\nor x^2 + 2ax + 2bx + a^2 + 2ab + b^2 - [x^2 + x^2 + 2ax + a^2 - x^2 - 2bx - b^2] = 0\nor 4bx + 2ab + 2b^2 = 0\nor 2b(2x+a+b) = 0\nor  2x + (a + b) = 0\nor 2x = -(a+b)\nTherefore x = -(a+b)/2\n\nSecond equation:\n(x+a)^2=(x+b)^2\nor (x+a)^2 - (x+b)^2 = 0\nor (x+a+x+b)(x+a-x-b) = 0\nor (2x+a+b)(a-b) = 0\nIf a = b then the equation becomes identity. So a is not equal to b.\nHence 2x + a + b = 0 \nor x = -(a+b)/2\n\nThus from both equations we have x = -(a+b)/2
15859
Variables in math not only represent numbers but any object. For instance, you can define a function f such as : f(x)=a*x\na and x are variables. \n\nBut you can also have equations where the variables are not numbers but functions (like differential equations).\nFor instance, find a function f that verifies :\nf+f'=1\n\nBasically, a variable is an object which can have different values, and the interest of a problem is to find the properties of an other object, depending of the values of this variables.\n\nA very simple example might be :\n\nLet's a and b be two numbers. How can a and b be chosen so that the equation : "a*x+b=0" has one or more solutions.\nHere, a and b are variables.
692871
It means that the equation can not be linear. If a=0, you have a straight line. Because a can not equal 0, you must end up with a parabola. Thus this equation HAS to be of the second order.
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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.
118612
Kerplode is right - but that's kind of complicated.  Basically you solve both in exactly the same way.  The only difference is that instead of an = you have an > or <.  \n\nIt just looks different, there really is no difference in how you solve them.
1102588
Nonlinear is a term used to relate one property to another. If the two properties are related so that a change in one property causes the other to change proportionally, they are linear. One way of looking at this is if you were to graph one versus the other, you would get a straight line with the slope being the ratio of the rate of change of one compared to the other.\n\nFor example, solids, liquids and gases tend to expand proportionally to temperature.\n\nProperties are nonlinear if they are not proportional to each other. For example, if one varies as the square root of the other, or the second or third power of the other, they are nonlinear.\nAir has many properties that are nonlinear. For example, the speed of sound in air is nonlinear when compared to the air temperature.
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The number of atoms/ions of an element on either side of the equation.\nThe total charge on either side of the equation.\nThe elements present are the same on either side of the equation.
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In a diagram where x denotes the horizontal axis and y the vertical axis, linear equations with only one variable (x or y) is either a horizontal or vertical line.\n\nexample 1:\n\ny = 3 is a horizontal line that connects all points in the diagram which have the y-value 3.\n\nexample 2:\n\nx = 5 is a vertical line that connects all points in the diagram which have the x-value 5.\n\nI hope this is clear?
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A linear equation looks like a line when it is graphed.  The standard form of the equation is y=mx+b where m is the slope, and b is the value of y at the point where the line crosses the y-axis.\n\nA non-linear equation does not look like a line.  It may look like a curve.  A non-linear equation could possibly be a parabola, a circle, a wave, a hyperbola, etc... but not a line!
12581
A first-order differential equation of the form:\n\na1(x)*(dy/dx) + a0(x)*y = g(x)\n\nis a linear equation.\n\nWhen g(x) = 0, it's homogenous.\n\nSo, (x^2-9)(dy/dx) + xy = 3 is linear and similarly, (x^2-9)(dy/dx) + xy = 0 is homogenous.\n\nA first-order differntial equation of the form\n\ndy/dx = g(x)h(y)\n\nis separable.\n\nSo dy/dx = (y^2)x(e^(3x+4y)) is seperable since (y^2)x(e^(3x+4y)) = (x(e^3x))((y^2)(e^4y)), and dy/dx = y + sin(x) is NOT separble since it cannot be defined as a product of two equations seperately of x and y.\n\nI can't recall how to discern if a diff equation is exact, but this webpage\n\nhttp://www.utpb.edu/scimath/wkfield/mod2/Exact.html\n\ngives a very detailed explanation.\n\nHope that helps.
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Yes, linear functions have limits.\n\nFor example, the limit of 3x+2 as x approaches 4 is 14. Here's how to prove it:\n\nFor every e>0, choose d=e/3. Then, if 0<|x-4|<d, we have\n|(3x+2)-14|=|3x-12|=3|x-4|\n<3d=e.\n\nFor this situation, we say that the limit of a function f(x) as x p&#92;approaches a is L is for every tolerance e>0, there is a tolerance d>0 so that whenever 0<|x-a|<d, we have\n|f(x)-L|<e.