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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 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 .
Interactive Graph: Example of a Linear Equation
Graph showing an example of two linear equations,
Key Term Reference
 constant
 Appears in these related concepts: Inverse Variation, Quadratic Functions of the Form f(x) = a(xh)^2 + k, and Direct Variation
 distance
 Appears in these related concepts: Inequalities with Absolute Value, Symmetry, and The Distance Formula and Midpoints of Segments
 equation
 Appears in these related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 function
 Appears in these related concepts: Inverse Functions, Solving Differential Equations, and Modal Mixture
 infinite
 Appears in these related concepts: Arithmetic Sequences, Sequences of Statements, and Summing Terms in an Arithmetic Sequence
 linear
 Appears in these related concepts: Trinomials of the Form ax^2 + bx + c, Where a is Not Equal to 1, Exponential Growth and Decay, and Graphs of Linear Inequalities
 point
 Appears in these related concepts: The Intermediate Value Theorem, Quadratic Functions of the Form f(x) = ax^2 + bx + c, Where a is not Equal to 0, and Circles
 term
 Appears in these related concepts: The Executive Departments, Basics of Graphing Polynomial Functions, and The 22nd Amendment
 unknown
 Appears in these related concepts: Graphing Equations, Solving Systems of Equations Using Matrices, and Linear Inequalities
 variable
 Appears in these related concepts: Related Rates, Math Review, and Psychology and the Scientific Method: From Theory to Conclusion
Sources
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Cite This Source
Source: Boundless. “Linear Equations and Their Applications.” Boundless Algebra. Boundless, 21 Jul. 2015. Retrieved 21 Jul. 2015 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/functionsequationsandinequalities3/linearequationsandfunctions22/linearequationsandtheirapplications1215519/