Systems of Equations

A system of equations is two or more equations with the same set of variables. By trying to figure out a system of equations, you have to find values for each of the unknown variables and discover how they fit in to every equation in the system. The solution of a system of linear equations with two variables is found by understanding how an ordered pair equals each equation in the system. If an ordered pair is a solution, it will make both equations true. System of equations can be expressed as word problems or as a numeric problem.

The systems of equations history go far back to ancient advanced cultures that used systems of equations to solve day-to-day problems. The first documented use of a system of equations appeared in the book Nine Chapters of Mathematical Art, which was created during the Han Dynasty, 2000 years ago. In 1750, a well-acknowledged mathematician, Gabriel Cramer made the well-known rule for solving systems of linear equations that features his name. The rule is a formula that solves systems of equations.

Overview

The Nine Chapters on the Mathematical Art is a book of 246 problems and solutions, most of which relate to day-to-day situations such as land measurement and construction. It contains pages that deal with field measurement, proportions, three-dimensional objects, rectangular arrays, and triangles. The Chinese used a variety of tools to make mathematical calculations. Bamboo counting rods, for example, were used as a way to give numbers place value.

Descartes was the first person to use a coordinate system to visualize algebraic equations. By doing so, he opened a pathway for many innovations relating to the coordinate system. Descartes (1596–1650) was a French philosopher, mathematician, and writer who made many contributions to the the field of analytical geometry. His book La Geometrie proposed the idea of making geometry and algebra into a single subject.

Gabriel Cramer was born on July 31, 1704 in Geneva, Switzerland. Cramer died on January 4th, 1752, in Bagnols-sur-Ceze, France. When he was only 18, he received his doctorate degree after submitting a thesis on the theory of sound. Cramer taught geometry and mechanics at the Academie de Clavin in Geneva, and he took his teaching to different countries and cities of Europe. Cramer made major contributions in literature and mathematics. His most famous piece of writing was Introduction à l'analyse des lignes courbes algébraique. Cramer modeled this book on Isaac Newton’s work on cubic curves.

Terms having to do with system of equations include inequality, linear equation, ordered pair, slope, and solution of an equation. An inequality is the relation between two expressions that are not equal. A linear equation is an equation between two variables that gives a straight line when plotted on a graph. An ordered pair is a pair of elements. Slope is the rise and run of an equation. The solution of an equation is a value that can be substituted for a variable in the equation that makes the equation true.

The solution is where the equations cross or intersect. There can be zero solutions, one solution or infinite solutions. When it is one solution, lines intersect one time. There is no solution when lines are parallel. There are infinite solutions when the equations make the same line.

Elimination

When doing the elimination method either use addition or subtraction to get an equation in one variable. When the coefficients of one variable are not the same, add the equations to eliminate a variable. When the coefficient of one variable is the same, subtract the equations to eliminate a variable. An example is

First write the system so that like terms are aligned. Eliminate one of the variables and solve for the other one. Substitute the value of the variable into one of the original equations and solve for the other variable. Lastly, write the answer as an ordered pair and check.

Substitution

The method of solving the equation using substitution works by figuring out one of the equations for one of the variables, and then putting this back into the other equation, substituting for the chosen variable and solving for the other. The first variable can then be solved. Think of it as "grabbing" what one variable equals from one equation and putting it into the other equation.

Lastly, write the resulting values as an ordered pair and check.

Word Problems

Many problems lend themselves to being solved with systems of equations. In real life, these problems can be incredibly complex. The key to solving the word problem is to identify the variables and construct two equations based on the information given. For example, Paul is responsible for buying a week's supply of food and medication for the dogs and cats at a local shelter. The food and medication for each dog costs twice as much as the supplies for a cat. He needs to feed 164 cats and 24 dogs. His budget is $4,240. How much can Paul spend on each dog for food and medication?

Graphing

When solving systems, intersections of lines are found. For two-variable systems, there are three possible types of solutions.

The graph in Figure 1 shows two crossing lines. This is called an independent system of equations. A consistent system of equations has one solution set for the different variables in the system or infinitely many sets of solution. In other words, as long as there is a solution, that system is referred to as being consistent. For the system to be consistent, two lines have to intersect at one point.

Figure 2 shows two lines that are parallel. Since the lines are parallel, there can be no solution. This is called an inconsistent system of equations. Inconsistent systems of equations are referred as such because there is no set of solutions for the system of equations. Inconsistent systems come about when the lines formed from the systems don't meet at any point.

The graph in Figure 3 appears to have only one line. It is, however, the same line drawn twice. These two lines, being the same line, cross at every point along their length. This is called a dependent system, and the solution is the whole line.

Bibliography

Aigner, Martin, and Gunter M. Ziegler. Proofs from the Book. New York: Springer, 2014.

Derbyshire, John. Unknown Quantity: A Real and Imaginary History of Algebra. New York: Plume, 2007.

Hanna, Gila. Explanation and Proof in Mathematics. New York: Springer, 2014.

Katz, Victor, and Karen Hunger Parshall. Taming the Unknown: A History of Algebra. Princeton: Princeton UP, 2014.

Millman, Richard, Peter Shiue, and Eric Brendan Kahn. Problems and Proofs in Numbers and Algebra. New York: Springer, 2015.

Polya, G. How to Solve It. Princeton: Princeton UP, 2014.

Strogatz, Steven. The Joy of X. New York: Mariner, 2013.