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Insights Into Algebra 1 - Teaching For Learning
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Workshop 3 Systems of Equations and Inequalities Topic Overview
Topic Overview:

Part 1: Systems of Equations

Part 2: Systems of Inequalities
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NCTM Standards

Topic overview
Part 1: Systems of Equations

A system of equations involves the relationship between two or more functions and can be used to model a number of real-world situations.

Mathematical Definition
Role in the Curriculum
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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
Mathematical Definition

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.)

Role in the Curriculum

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.

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Next: Part 2: Systems of Inequalities
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