## Solving Systems of Equations Graphically

### Johnny WolfeSanta Rosa District Schools

#### Description

Students solve a system of equations by graphing.

#### Objectives

Understands the relative size of integers, rational numbers, irrational numbers, real numbers and complex numbers.

Understands that numbers can be represented in a variety of equivalent forms using integers, fractions, decimals, and percents, scientific notation, exponents, radicals, absolute value, or logarithms.

Understands and explains the effects of addition, subtraction, multiplication and division on real numbers, including square roots, exponents, and appropriate inverse relationships.

The student identifies and plots ordered pairs in all four quadrants of a rectangular coordinate system (graph) and applies simple properties of lines.

Using a rectangular coordinate system (graph), applies and algebraically verifies properties of two-and three- dimensional figures, including distance, midpoint, slope, parallelism, and perpendicularity.

#### Materials

- Overhead transparencies (if examples are to be worked on overhead) for Solving Systems of Equations Graphically (see attached file).

- Graph Paper

- Solving Systems of Equations Graphically Examples (see attached file).

- Solving Systems of Equations Graphically Worksheet (see attached file).

- Solving Systems of Equations Graphically Checklist (see attached file).

#### Preparations

1. Prepare transparencies (if teacher uses overhead for examples) for Solving Systems of Equations Graphically Examples (see attached file).

2. Have marking pens (for overhead).

3. Have graph paper.

4. Have Solving Systems of Equations Graphically Examples (see attached file) prepared and ready to demonstrate to students.

5. Have enough copies of Solving Systems of Equations Graphically Worksheet (see attached file) for each student.

6. Have enough copies of Solving Systems of Equations Graphically Checklist (see attached file) for each student.

#### Procedures

Prior Knowledge: Students should be familiar with basic operation skills such as addition, subtraction, multiplication, division, exponents, fractions, decimals, solving equations, and point-slope form of a linear equation. Note: This lesson does not address the following: irrational, real, and complex numbers; decimals, percents, scientific notation, exponents, radicals, absolute value, logarithms; square roots, exponents, and appropriate inverse relationships; distance, midpoint, perpendicularity. This lesson contains a checklist to assist the teacher in determining which students need remediation. The sole purpose of this checklist is to aide the teacher in identifying students that need remediation. Students who meet the “C” criteria are ready for the next level of learning.

1. Give students an example of how a system of equations is used in everyday situations (see # 1 on attached file Solving Systems of Equations Graphically Examples). Answer student questions and comments.

2. Help students set up a system of equations for # 1 above (see # 2 on attached file Solving Systems of Equations Graphically Examples). Answer student questions and comments.

3. Aide students in graphing the system of equations from # 2 above. Discuss what the intersection between the lines represents (see # 3 on attached file Solving Systems of Equations Graphically Examples). Answer student questions and comments.

4. Discuss with students what the term “system of equations” represents and the solution can be written as an ordered pair (see # 4 on attached file Solving Systems of Equations Graphically Examples). Answer student questions and comments.

5. Work example #5 (see attached file Solving Systems of Equations Graphically Examples). Answer student questions and comments.

6. Work example #6 (see attached file Solving Systems of Equations Graphically Examples). Answer student questions and comments.

7. Work example #7 (see attached file Solving Systems of Equations Graphically Examples). Answer student questions and comments.

8. Discuss the terms “consistent,” “independent,” “dependent,” and “consistent” with the students (see # 8 on attached file Solving Systems of Equations Graphically Examples). Answer student questions and comments.

9. Work example #9 (see attached file Solving Systems of Equations Graphically Examples). Answer student questions and comments.

10. Review with students the possibilities for graphs when using two linear equations in two variables (see # 10 on attached file Solving Systems of Equations Graphically Examples). Answer student questions and comments.

11. Work example #11 (see attached file Solving Systems of Equations Graphically Examples). Answer student questions and comments.

12. Work example #12 (see attached file Solving Systems of Equations Graphically Examples). Answer student questions and comments.

13. Work example #13 (see attached file Solving Systems of Equations Graphically Examples). Answer student questions and comments.

14. Aide students in reading and interpreting a graph in which the miles per gallon of various automobiles is compared (see # 14 on attached file Solving Systems of Equations Graphically Examples). Answer student questions and comments.

15. Work example #15 (see attached file Solving Systems of Equations Graphically Examples). Answer student questions and comments.

16. Distribute graph paper.

17. Distribute the Solving Systems of Equations Graphically Worksheet (see attached file).

18. Distribute the Solving Systems of Equations Graphically Checklist (see attached file). Describe what constitutes an “A,” “B,” “C,” “D,” and an “F” in the CHECKLIST. Explain that this checklist is they will be assessed. This will help the teacher see in what they need assistance and remediation.

19. The student will write their response on the Solving Systems of Equations Graphically Worksheet.

20. The teacher will move from student to student observing the students work and lending assistance. Offer feedback that is specific in praise and instruction.

#### Assessments

Student worksheets will be taken up and scored according to “Solving Systems of Equations Graphically Checklist”. These scores may be placed in the grade book if students have been given sufficient practice to learn the material. If this is the first time students have been introduced to the material, this would be a formative assessment activity.

#### Extensions

Have students do some research on post-high school education and non-post-high school education. Have them determine what their salary would be after graduation in they did not pursue further education and what the projected salary would be in 10 years and 20 years. Also have the students find the same information for a career that requires a college degree. Let students draw graphs for both equations and determine the point of intersection.