LINEAR EQUATION
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.
One variable
A linear equation in one unknownx may always be rewritten
If a ≠ 0, there is a unique solution
If a = 0, then, when b = 0 every number is a solution of the equation, but if b ≠ 0 there are no solutions (and the equation is said to be inconsistent.)
Two variables
A common form of a linear equation in the two variables x and y is
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 determines the slope or gradient of that line, and the constant term b 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, x2, y1/3, and sin(x) are nonlinear.
Applications of Linear Equations
Consider, for example, a situation in which one has 45 feet of wood to use for making a bookcase. If the height and width are to be 10 feet and 5 feet, respectively, how many shelves can be made between the top and bottom of the frame?
To solve this equation, we can use a linear relationship:
Nv+Mh=45
where v and h respectively represent the length in feet of vertical and horizontal sections of wood. N and M represent the number of vertical and horizontal pieces, respectively. Knowing that there will be only two vertical pieces, this formula can be simplified to:
2⋅10+M⋅5=45
Solving for M, we find that there is enough material for 5 shelves (3 shelves if you don't count the top and bottom).
Similarly, we can use linear equations to solve for the original price of an item that is on sale. For example, consider an item that costs $24 when on a 40% discount. If the original price is x, we can write the following relationship:
x−0.4⋅x=24
Solving for x, we find that the original price was $40.
Using similar models we can solve equations pertaining to distance, speed, and time (Distance=Speed*Time); density (Density=Mass/Volume); and any other relationship in which all variables are first order. For example,imagine these linear equations represent the trajectories of two vehicles. If the drivers want to designate a meeting point, they can algebraically find the point of intersection of the two functions, as seen in .
Source: Boundless. “Linear Equations and Their Applications.” Boundless Algebra. Boundless, 21 Jul. 2015. Retrieved 10 Dec. 2015 from www.boundless.com/algebra/textbooks/boundless-algebra-textbook/functions-equations-and-inequalities-3/linear-equations-and-functions-22/linear-equations-and-their-applications-121-5519/
