A linear equation is an algebraic equation that is of the first order—that is, an equation in which each term is either a constant or the product of a constant and a variable raised to the first power.
Linear equations are commonly seen in two dimensions, but can be represented with three, four, or more variables. There is in fact a field of mathematics known as linear algebra, in which linear equations in up to an infinite number of variables are studied.
Linear equations can therefore be expressed in general (standard) form as:
where a, b, c, and d are constants and x, y, and z are variables. Note that there can be infinitely more terms. This is known as general (or standard) form.
Applications of Linear Equations
Linear equations can be used to solve many problems, both everyday and technically specific.
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:
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:
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:
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 (Figure 1).