Systems of Equations

Overview of Objectives, students should be able to:

1. Determine if a given ordered pair is a solution to a
system of linear equations.
2. Solve systems of linear equations using graphing .
3. Solve systems of linear equations by substitution.
4. Solve systems of linear equations by addition.
5. Select the most efficient method for solving a system
of linear equations.
6. Identify systems that have no solution or infinitely
many solutions.

Main Overarching Questions:

1. When is an ordered pair a solution of a system?

2. How do you solve systems of equations by graphing?

3. How do you solve systems of equations by algebraic methods ?

4. Compare and contrast the methods of solving systems for efficiency and accuracy.

5. How do you determine when a system has no solution or infinitely many solutions?

Objectives:

Activities and Questions to ask students:

• Determine if a given ordered pair is a solution to a
system of linear equations.

1. Present two linear equations and have students substitute an ordered pair into x and y.
What is meant by the term “solution to an equation”? Does the ordered pair create true
or false statements? Is this point a solution for both of these equations?

. When is an ordered a solution of a system of linear equations?

• Solve systems of linear equations using graphing.

Group activity or teacher directed:

1. Students will graph 3 systems of linear equations: a) a pair of intersecting lines b) a pair of
parallel lines c) two equations that are the same line ( coinciding lines according to some
texts)

2. Direct students to compare the three systems: a) describe the type of lines b)describe how
many points they have in common c ) compare the equations within each system

3. Groups may present their results.

4. Summarize using 3‐column note format.

• Solve systems of linear equations by substitution.

1. What does substitution mean? What can be substituted without changing the solution of an
equation?

2. If 2 equations are solved for y like y = 2x – 1 and y = x + 4, can you say that 2x‐1 = x + 4. Why or why not? Now that you can solve for x, how can you find y?

For students who struggle with substitution method, try the above method.

3. Demonstrate technique of solving one equation for a variable. How can we use
substitution to combine these two equations into a single equation with one variable?

4. How does this help us solve the system? How do you find the second variable?

5. What happens if you substitute into the wrong variable?

• Solve systems of linear equations by addition

1. How do you add two equations ? What parts can you add?

2. Ask students to add two given equations (where a variable will cancel). How does this help
us solve the system? How do you find the second variable?

3. Ask students to add two given equations where a variable does not cancel? Does this help us
solve the system? Why not? What needs to happen? What is it about the coefficients that
makes a variable cancel when adding?

4. How can we change an equation so that a variable will cancel when we add? IF we change
one part of the equation, what must be done to the remaining terms in the equation ?

5. Introduce or reemphasize term: equivalent equations

• Select the most efficient method for solving a system
of linear equations.

• Identify systems that have no solution or infinitely
many solutions.

1. Have students compare/contrast the 3 methods of solving a system. Do lines always
intersect at integer points? Can you always read the coordinates of the intersection on a
graph? Can you determine fraction solutions when solving using algebraic methods.

2. Ask students to make a conclusion about the efficiency and accuracy of each method.

3. Give students a system with no solution and ask them to use either the substitution or
addition method to solve. What happens to the variables? What kind of statement is left? Is
it true or false? Ask students to solve each equation for y and compare the slopes and yintercepts.
What kind of lines are they and how many solutions are there for this system?

4. Give students a system with infinitely many solutions and ask them to solve using addition or
substitution method. What happens to the variables and the constants? What statement is
left? True or false? Ask students to solve both equations for y and to make a conclusion
about the type of lines and the number of solutions .