# Linear Equation - Not always a line

Definition: A linear equation is an equation in which each term is either a constant or the product of a constant and a variable.

Examples of linear equations: x=3    2y+5=2    y=2x-1    3x+2y=5    s=4t-3    x+2y-3z-5=0
 Regulations: The graph of a linear equation is a line if it has exactly two variables. 1 variable -> a point 2 variables -> a line 3 variables -> a plane 4 plus variables has no physical representation! See!
 Linear Equation # Variables Simplified Form Graph x=3 1 ok Point on Number Line This browser does not have a Java Plug-in. Get the latest Java Plug-in here. 2y+5=2 1 y=-1.5 Point on Number Line This browser does not have a Java Plug-in. Get the latest Java Plug-in here. 14=2t 1 t=7 Point on Number Line This browser does not have a Java Plug-in. Get the latest Java Plug-in here. y=2x-1 2 y=2x-1 slope-intercept 2x-y=1 standard Line in 2D Plane 3x+2y=5 2 3x+2y=5 standard y=-1.5x+2.5 slope-intercept Line in 2D Plane 2s+4=8t-2 2 s=4t-3 slope-intercept 4t-s=3 standard Line in 2D Plane z=2y-x+2 3 x-2y+z=2 standard form Plane passes through: Point on x-axis   (2,0,0) Point on y-axis   (0,-1,0) Point on z-axis   (0,0,2) Plane in 3D Space A line in 3D space is the intersection of 2 planes - that is, the solution of a system of 2 linear equations in 3 variables?. 2x_1+\sqrt{3}x_2-3x_3+0.5x_4=\pi 4 ok no physical representation Notice: 1 variable: point on line, 2 variables: line in plane, 3 variables: plane in space.
Regulations: A linear function is a linear equation in two variables. (The graph of a linear function is a line.)
Is it a linear equation?    Examples
Regulation a: All the variables in a linear equation are "on top" (in the numerator of any fraction).
Regulation b: No two variables in a linear equation are "together" (multiplied).
Regulation c: A linear equation has no exponents (except 1) or roots on the variables.
Regulation d: A linear equation is an equation.
 Algebra Linear Equation? y=\frac{4}{x} No, the variable x is in the denominator. a 3xy=4 No, the variables x and y are multiplied. b ax+by=c Yes, usually the letters a , b and c represent constants (numbers) and not variables. Teachers usually use letters from the beginning of the alphabet to mean constants A, B, C, D, .... and a, b, c, d, ... and letters from the end of the alphabet to be variables. When in doubt - ASK! Your teacher must be clear on this or it isn't fair! 2x+y-4 No, this is an expression - NOT an equation. d x^2=4 No, the variable x has an exponent of 2. c \sqrt{2}t-\frac{3w}{2}=4 Yes, it doesn't matter that the numbers are ugly (rationals and irrationals), the variables t and w are "clean". \sqrt{x}-\frac{3y}{2}=4 No, the variable x is inside a square root. c \frac{2}{x}=1 Mathematically speaking No, because the variable x is in the denominator. BUT, your teacher may say it is linear, because it can be made into a linear equation by multiplying both sides by x . This is different than example 1 because that example is not linear no matter what.

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