Systems of Equations and Inequalities

Explanation

A system of equations consists of two or more equations that have variables that represent the same items. For example, the equations 2x + 3y = 4 and 3x + 4y = 5 form a system if x represents the same thing in both equations, y represents the same thing in both equations, and both equations refer to the same context. In order to “solve the system,” students must find values for the variables that make both statements true.

Solving a system of equations can be useful in calculating the cost difference between various payment plans, or in figuring out when a business enterprise will break even. Examining the cost of video rentals at two different stores as a function of the number of videos rented, or looking at the monthly cost of two cell phone plans as a function of minutes used are examples of systems of equations.

To solve a system means to find the x- and y-values for which both of the equations are true. Systems of linear equations can be solved using a variety of methods. One method is to graph the equations as two lines and examine them. If the lines intersect at exactly one point, there is one solution to the system, and the system is called consistent. If the two lines are on top of one another, there are an infinite number of solutions, because each point on the line is considered a solution (this system is called dependent). If the two lines are parallel, there is no solution (this system is called inconsistent).

Additional methods of solving the system include substitution, linear combinations, determinants (Cramer’s Rule), and matrices. In Algebra 1, students are usually taught the graphing method, the substitution method, and the linear combination method.

Other examples:

• Car A costs $25,000 to purchase and 35 cents a mile to drive. Car B costs $18,500 to purchase and 40 cents a mile to drive. How many miles (x) would one have to drive before the total cost (y) of driving Car A and Car B would be the same? This system involves the following equations:25,000 + .35x = y
  18,500 + .40x = y
  • The revenue for a certain business is $2,500 per day. The costs for the same business consist of a fixed amount of $10,000 and a flexible amount of $1,000 per day. How many days will the business need to operate before revenue equals cost? This system involves the following equations:10,000 + 1,000x = y
    2,500x = y

Two linear functions with the same variables form a system of equations.

Also known as simultaneous equations, a system of equations consists of two or more equations that are conditions imposed simultaneously on all of the variables, but may or may not have common solutions. For example,
x + y = 2 and 3x + 2y = 5, when treated as simultaneous equations, are satisfied by x = 1, y = 1, these values being the coordinates of the point of intersection of the straight lines which are the graphs of the two equations. Simultaneous linear equations are simultaneous equations, which are linear (of the first degree) in the variables.
(Source: James, Robert C. and Glenn James. Mathematics Dictionary (4th edition). New York: Chapman & Hall, 1976.)

According to the National Council of Teachers of Mathematics (NCTM), the study of mathematical patterns and relationships in the middle grades should focus on patterns that relate to linear functions – that is, functions that arise when there is a constant rate of change. Students should solve problems in which they use tables, graphs, words, and symbolic expressions to represent these functions and patterns of change.

In the later grades, students have the opportunity to enlarge on these earlier studies. The NCTM states in Principles and Standards for School Mathematics (PSSM):

… students should build on their prior knowledge, learning more varied and more sophisticated problem-solving techniques. They should increase their abilities to visualize, describe, and analyze situations in mathematical terms. They need to learn to use a wide range of explicitly and recursively defined functions to model the world around them. Moreover, their understanding of the properties of those functions will give them insights into the phenomena being modeled.

For example, students in Algebra 1 should be able to recognize situations that use a system of equations; be able to write a system of equations from a given set of information; and be able to solve a system of equations problem using the substitution method, linear combination method, and graphing method. The following problem illustrates these processes.

Jenny mailed 30 postcards and letters. The bill at the post office was $9.28. The postcards cost $0.23 each to mail, and the letters cost $0.37 each. How many of each kind did she mail?

Step 1: Recognize that the problem situation involves a system of equations. Because there are two unknowns (number of postcards and number of letters), and two pieces of information about each (total items mailed and total cost), this information can be solved using a system of equations. We can let x = the number of postcards mailed, and y = the number of letters mailed.

Step 2: Be able to write a system of equations from a given set of information. We know how many total items were mailed, and the total cost. Each set of information can be written as an equation, namely x + y = 30, and 0.23x + 0.37y = 9.28.

To solve the system using the substitution method, first solve one of the equations for a given variable (either x or y). Then, substitute the expression into the other equation and solve it for the remaining variable.

y = -x + 30 (solved the first equation for y)
0.23x + 0.37(-x + 30) = 9.28 (substituted the expression -x + 30 into the second equation in place of y)
0.23x – 0.37x + 11.10 = 9.28
-0.14x = -1.82
x = 13

Find the value of y by substituting the value of x into either equation.
x + y = 30
13 + y = 30
y = 17.
State the solution to the problem. Jenny mailed 13 postcards and 17 letters.

Another way to solve this problem is to use the linear combination method. The approach to this method is to multiply one of the equations by a constant so that when the two equations are subsequently added together, one of the two variables is eliminated. In the example below, multiply the first equation by -0.23

x + y = 30
0.23x + 0.37y = 9.28

-0.23x + (-0.23y) = -6.90
0.23x + 0.37y = 9.28

Adding the two equations together produces 0.14y = 2.38, and y = 17. Find the value of x by substituting the value of y into either equation.
x + y = 30
x + 17 = 30
x = 13.
State the solution to the problem. Jenny mailed 13 postcards and 17 letters.

The third way to solve this problem is by graphing each of the equations in the system and finding the point where the two lines intersect. Students can graph the equations on graph paper, or use a graphing calculator to find the graphs. Either way, the student will usually first solve each equation for y so that it is written in the slope-intercept form.

x + y = 30
0.23x + 0.37y = 9.28

y = -x + 30

State the solution to the problem. The two lines intersect at the point (13, 17). Jenny mailed 13 postcards and 17 letters.

Students should understand the meaning of, and solutions to, systems of linear equations by the end of Algebra 1. In later study, students will solve systems of non-linear equations. The principles students learn solving systems of linear equations helps them understand the process of solving more complicated systems of equations.

Explanation

A system of linear inequalities is an extension of a system of linear equations and consists of two (or more) linear inequalities that have the same variables. For example, 2x + 3y < 4 and 3x + 4y < 5 constitute a system of inequalities if x represents the same item in both equations, y represents the same item in both equations, and both equations describe the same context. The solution consists of a region in the coordinate plane that satisfies all of the inequalities in the system.

Systems of inequalities can be useful in determining which combinations of products sold by a business will yield the maximum profit. For instance, one can find the combination of paintings, given a set of constraints, that produces the greatest profit for a painter who sells two different types of artwork.

To solve a system of linear inequalities, one must first determine the boundary lines by graphing each inequality as though it were an equation and then identifying the region where all of the inequalities would be shaded at the same time. In general, if the inequality is “less than,” one shades below the line, and if the inequality is “greater than,” one shades above the line. The shaded region is called the feasible region because it represents all the possible points that satisfy the system of inequalities.

One (and sometimes more than one) point in the feasible region is considered the optimum point. This is the point where profits are maximized or costs minimized. The optimum point is located on a corner of the feasible region (or the intersection of two of the boundary lines), and its coordinates are usually integer values.

To visualize what this entire process looks like, go to the Role in the Curriculum Section, below, and follow through the example.

Other examples:

• A school system wants to hire a combination of teachers and aides. Given a set of constraints, what combination of teachers and aides would be least costly?
• A pet store sells dogs and cats. Given a set of constraints, what combination of dogs and cats should they sell to maximize their profit?

Two or more linear inequalities with the same variables form a system of inequalities.

Also known as simultaneous inequalities, a system of inequalities consists of two or more inequalities that are conditions imposed simultaneously on all the variables, but may or may not have common solutions.

For example, x² + y² < 1 and y > 0, when treated as simultaneous inequalities, have as their solution set the set of all points above the x-intercept and inside the unit circle about the origin. The interior of a convex polygon is the graph (or solution set) of suitable simultaneous linear inequalities – in two variables for the polygon. (Source: James, Robert C. and Glenn James. Mathematics Dictionary (4th edition). New York: Chapman & Hall, 1978.)

Linear programming is an important element in solving systems of inequalities. A linear programming problem is an optimization problem for which:

1. The function to be maximized or minimized – called the objective function – is a linear function of the decision variables.
2. The values of the decision variables must satisfy a set of constraints. Each constraint must be a linear equation or a linear inequality.
3. The variables must be non-negative.

Another widely accepted definition of linear programming is:

The mathematical theory of the minimization or maximization of a linear function subject to linear constraints. As often formulated, it is the problem of minimizing a linear expression in two or more variables subject to one or more linear constraints. (Source: James, Robert C. and Glenn James. Mathematics Dictionary (4th edition). New York: Chapman & Hall, 1978.)

Students should begin to work with systems of linear inequalities and then investigate a broad array of linear programming problems. This is an opportunity to explore more sophisticated mathematical modeling situations. Modeling involves identifying and selecting relevant features of a real-world situation, representing those features symbolically, and analyzing and considering the accuracy and limitations of the model. Linear programming problems require all of the modeling processes listed above. They provide students with a rich opportunity to glimpse important applications that are used in a wide range of business settings.

For example, in Algebra 1, students should be able to recognize situations that use a system of inequalities, write a system of inequalities from a given set of information or constraints, and solve a system of inequalities by graphing each of the inequalities on the same grid and determining the region (if any) which satisfies all of the inequalities. Algebra 1 curricula usually ensure that students have experience graphing systems of linear inequalities by posing a variety of linear programming problems. These problems use systems of linear inequalities as a part of their solution. After students have solved the system of linear inequalities, they can look for an optimum point that produces either a maximum profit or a minimum cost. In linear programming, the solution to a system of linear inequalities is called the feasible region. The following example illustrates these processes.

A manufacturer of skis produces two types: telemark and cross-country. It takes the manufacturer 4 hours to produce each pair of telemark skis and 2 hours to produce each pair of cross-country skis. The maximum time available for production each week is 80 hours. It takes 2 hours to wax and put finishing touches on each pair of telemark skis and it takes 2 hours to complete the same processes for the cross-country skis. The maximum time allowed for waxing and finishing altogether is 64 hours each week. When the skis are sold the manufacturer makes $140 profit on the telemark skis and $100 profit on the cross-country skis. How many skis of each type must be produced each week to achieve a maximum profit?

Step 1: Identify the variables.
Let x = the number of pairs of telemark skis manufactured each week.
Let y = the number of pairs of cross-country skis manufactured each week.

Step 2: Write the inequalities based on the given constraints.
Production time constraint:
Finishing time constraint:   
Two other constraints that are implied are and because it isn’t possible to produce a negative number of skis.

Step 3: Write the profit or cost equation, also known as the objective function.
Profit or Objective equation: P = 140x + 100y

Step 4: Graph the system of inequalities to find the feasible region. To graph the inequalities, solve each inequality for y so that it is written in slope-intercept form. An alternate method is to find the x-intercept and y-intercept of each linear inequality by first substituting y = 0, and then x = 0 into each equation. This is the technique that Patricia’s students used in the video for Workshop 3, Part II. As shown by the graphing calculator images below, the feasible region is the region where both inequalities are shaded at the same time. It is the region between the x-intercept, y-axis, and below the two lines. Each point in the feasible region represents a possible combination of telemark skis and cross-country skis that the manufacturer can produce and satisfy both constraints.

Step 5: Identify the corner points in the feasible region. There are four corner points in the feasible region: (0, 0) (0, 32) (20, 0) and (8, 24).

Step 6: Substitute the values of each of the corner points into the objective function, and identify either the maximum profit, or the minimum cost. In this particular example, we are looking for the maximum profit.

Step 7: State the solution to the problem. To attain the highest possible profit, the manufacturer should produce eight pairs of telemark skis and 24 pairs of cross-country skis each week, for a profit of $3520.

An understanding of the meaning of and how to solve systems of linear inequalities should be accomplished at the end of Algebra 1. In later study, students will solve systems of non-linear inequalities. The principles students learn solving systems of linear equations help them understand the process of solving more complicated systems of equations. Students might also spend time studying more difficult linear programming problems, including those involving three unknowns.

The links below are to pages within stable sites and are current as of the date of publication of this workshop. Due to the ever-changing nature of the Web, it is possible that some links may change. Should you reach a non-working link, we recommend entering a couple words from its description into the site’s search function, or into a Web-based search engine.

Related Standards:

NCTM Middle Grades Algebra Standard
http://standards.nctm.org/document/chapter6/alg.htm
This Web page describes what students should know and be able to do algebraically in grades 6-8, and offers suggestions for the type of classroom activities necessary to develop conceptual understanding.

NCTM High School Algebra Standard
http://standards.nctm.org/document/chapter7/alg.htm
This Web page describes what students should know and be able to do algebraically in grades 9-12, and offers suggestions for the type of classroom activities necessary to develop conceptual understanding.

Systems of Equations, Systems of Inequalities, and Linear Programming:

Understanding Distance, Speed, and Time Relationships Using Simulation Software
http://standards.nctm.org/document/eexamples/
Example 5.2 includes a software simulation of two runners on a track. Students can control the speeds and starting points of the runners, watch the race, and examine a graph of the time-versus-distance relationship. The computer simulation uses a context familiar to students, and the technology allows them to analyze the relationships more deeply by manipulating the environment and observing the changes that occur. This applet could be used as an introduction to solving systems of equations.

High School Operations Research
http://www.hsor.org/
High School Operations Research develops instructional materials for use in high school mathematics classrooms. This site has some examples of linear programming problems.

Figure This!
http://www.figurethis.org/index40.htm
The “Figure This!” site provides a growing set of challenges intended to promote students’ problem-solving ability. The site is designed to entice parents and other adults to do mathematics problems with middle school kids and thus understand the high-level mathematics problems that are part of the middle school curriculum. There are currently 80 challenges. In this particular challenge, students are presented with a diagram depicting the “cost” of various combinations of frowns, smiles, and neutral expressions on a face. They are asked to determine how much is a smile worth. In order to answer the question, students may either use the method of guess/check or rely on patterns to develop systems of equations.

Exploring Linear Functions: Representational Relationships
http://standards.nctm.org/document/eexamples/
Example 7.5 presents a series of explorations based on two linked representations of linear functions. The grades 9-12 section on the “Problem Solving Standard” also includes a helpful description of how a teacher engaged her students in problem solving and reasoning with tasks such as those presented on the site.

Quick Math
http://www.quickmath.com/
This Web site gives students access to a powerful online computer algebra system. It provides such algebraic manipulations as expanding, factoring, and simplifying expressions, as well as combining rational expressions and generating partial fractions. It can solve almost any equation or inequality a high school student will encounter, giving exact answers when possible and approximations in other cases. Matrix operations and beginning calculus are also within its grasp, as is graphing of two-dimensional functions and relations. There is almost nothing it can’t do, and the site responds to requests very quickly!

Building Understanding:

Olson, Steve and Susan Loucks-Horsley (eds). Inquiry and the National Science Education Standards: A Guide for Teaching and Learning. Committee on the Development of an Addendum to the National Science Education Standards on Scientific Inquiry, National Research Council; 2000.
This book deals with the challenges of teaching science and math, and is available for free online at http://www.nap.edu/

How People Learn: Brain, Mind, Experience, and School (Expanded Edition). Committee on Developments in the Science of Learning, 2000.
This book makes important connections between classroom learning and everyday activities. It contains additional material from the Committee on Learning Research and Educational Practice, National Research Council, and is available for free online at http://www.nap.edu/

Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools. Committee on Programs for Advanced Study of Mathematics and Science in American High Schools, National Research Council; 2002.
This book is available for free online at http://www.nap.edu/

Teaching English Language Learners:

TESOL (Teachers of English to Speakers of Other Languages)
http://www.tesol.org/index.html
This is a well-organized site with lots of curricular, practical, professional, and social information.

The National Association for Bilingual Education (NABE)
http://www.nabe.org/
The site provides access to the Bilingual Research Journal Online, NABE publication that focuses on issues in the bilingual education field. Users can also access NABE News, the Association’s official news magazine. Published articles and research papers provide further support and information.

Dave’s ESL Café
http://www.eslcafe.com/
One of the most popular sites for ESL teachers, Dave’s Café offers materials and ideas for the classroom, helpful resources, and opportunities for ESL teachers to communicate with one another.

Books

August, Diane and Kenji Hakuta (eds). Educating Language-Minority Children.Committee on Developing a Research Agenda on the Education of Limited-English-Proficient and Bilingual Students, National Research Council and Institute of Medicine; 1998.
This book is available for free online at http://www.nap.edu/

August, Diane and Kenji Hakuta (eds). Improving Schooling for Language-Minority Children: A Research Agenda. Committee on Developing a Research Agenda on the Education of Limited English Proficient and Bilingual Students, National Research Council and Institute of Medicine; 1997.
This book is available for free online at http://www.nap.edu/

Cummins, Jim. Empowering Minority Students. Sacramento: California Association for Bilingual Education, 1989.

Ladson-Billings, Gloria. “Culturally Relevant Teaching.” The College Board Review, 155 (1990): 20-25.

Tate, William F. “School Mathematics and African American Students: Thinking Seriously about Opportunity-to-Learn Standards.” Educational Administration Quarterly, 31 (1995): 424-48.

Changing the Faces of Mathematics: Perspectives on Multiculturalism and Gender Equity. Reston, VA: National Council of Teachers of Mathematics, 2000.
This book contains volumes of articles written by authors who are committed to the goals of the mathematics education reform movement. Each chapter offers a compelling narrative about a specific issue.