# Linear Equations and Their Applications

## Linear equations are those with one or more variables of the first order.

#### Key Points

• Linear equations can be expressed in the form: Ax+By+Cz+...=D.

• Linear equations can contain one or more variables; it's possible for such an equation to include an infinite number of variables.

• Linear equations can be used to solve for unknowns in any relationship in which all the variables are first order.

#### Terms

• A polynomial equation of the first degree (such as x = 2y - 7).

#### Figures

1. ##### Interactive Graph: Example of a Linear Equation

Graph showing an example of two linear equations, $y=-x+5$ (red) and $y=0.5x+2$ (blue). 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.

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:

$ax+by+cz+...=d$

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:

$Nv+Mh= 45$

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:

$2\cdot10+M\cdot5=45$

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\cdot 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 (Figure 1).

#### Key Term Glossary

algebraic
or function}} Containing only numbers, letters and arithmetic operators.
##### Appears in these related concepts:
constant
An identifier that is bound to an invariant value.
##### Appears in these related concepts:
distance
The amount of space between two points, measured along a straight line
##### Appears in these related concepts:
equation
An assertion that two expressions are equal, expressed by writing the two expressions separated by an equals sign. E.g. x=5.
##### Appears in these related concepts:
function
A relation in which each element of the input is associated with exactly one element of the output.
##### Appears in these related concepts:
infinite
Boundless, endless, without end or limits; innumerable.
##### Appears in these related concepts:
linear
Of or relating to a class of polynomial of the form y = ax + b .
##### Appears in these related concepts:
linear equation
A polynomial equation of the first degree (such as x = 2y - 7).
##### Appears in these related concepts:
Linear equation
A polynomial equation of the first degree (such as x = 2y - 7).
##### Appears in these related concepts:
point
An entity that has a location in space or on a plane, but has no extent
##### Appears in these related concepts:
term
any value (variable or constant) or expression separated from another term by a space or an appropriate character, in an overall expression or table.
##### Appears in these related concepts:
variable
A symbol that represents a quantity in a mathematical expression, as used in many sciences