Linear Equations and Their Applications
Linear equations are those with one or more variables of the first order.
Learning Objective

Apply linear equation models to the real world
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.
Term

linear equation
A polynomial equation of the first degree (such as x = 2y  7).
Full Text
A linear equation is an algebraic equation that is of the first orderâ€”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 .
Interactive Graph: Example of a Linear Equation
Graph showing an example of two linear equations,
Key Term Reference
 constant
 Appears in these related concepts: Graphing Quadratic Equations in Vertex Form, Inverse Variation, and Direct Variation
 distance
 Appears in these related concepts: Inequalities with Absolute Value, The Distance Formula and Midpoints of Segments, and Linear Mathematical Models
 equation
 Appears in these related concepts: Graphs of Equations as Graphs of Solutions, Equations and Inequalities, and What is an Equation?
 function
 Appears in these related concepts: Functions and Their Notation, The Vertical Line Test, and What is a Linear Function?
 infinite
 Appears in these related concepts: Arithmetic Sequences, Summing Terms in an Arithmetic Sequence, and Sequences of Mathematical Statements
 linear
 Appears in these related concepts: Exponential Growth and Decay, Graphs of Linear Inequalities, and Factoring General Quadratics
 point
 Appears in these related concepts: Polynomial and Rational Functions as Models, The Intermediate Value Theorem, and Graphing Quadratic Equations In Standard Form
 product
 Appears in these related concepts: Measuring Reaction Rates, Writing Chemical Equations, and Basic Operations
 term
 Appears in these related concepts: The 22nd Amendment, Basics of Graphing Polynomial Functions, and Democracy
 unknown
 Appears in these related concepts: Graphing Equations, Models Involving Nonlinear Systems of Equations, and Introduction to Variables
 variable
 Appears in these related concepts: Controlling for a Variable, Fundamentals of Statistics, and Math Review
Sources
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