A.4ef 
The student will solve multistep linear and quadratic equations in two variables, including
e) solving systems of two linear equations in two variables algebraically and graphically; and
f) solving realworld problems involving systems of equations.
Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions. 

Essential Knowledge and Skills 
 Given a system of two linear equations in two variables that has a unique solution, solve the system by substitution or elimination to find the ordered pair which satisfies both equations.
 Given a system of two linear equations in two variables that has a unique solution, solve the system graphically by identifying the point of intersection.
 Determine whether a system of two linear equations has one solution, no solution, or infinite solutions.
 Write a system of two linear equations that models a realworld situation.
 Interpret and determine the reasonableness of the algebraic or graphical solution of a system of two linear equations that models a realworld situation.

Essential Understandings 
 A system of linear equations with exactly one solution is characterized by the graphs of two lines whose intersection is a single point, and the coordinates of this point satisfy both equations.
 A system of two linear equations with no solution is characterized by the graphs of two lines that are parallel.
 A system of two linear equations having infinite solutions is characterized by two graphs that coincide (the graphs will appear to be the graph of one line), and the coordinates of all points on the line satisfy both equations.
 Systems of two linear equations can be used to model two realworld conditions that must be satisfied simultaneously.
 Systems of equations can be used as mathematical models for realworld situations.
 Set builder notation may be used to represent solution sets of equations.
 The product of the slopes of perpendicular lines is 1 unless one of the lines has an undefined slope.


Vertical Articulation 
 SOL 7.14  a) solve one and twostep linear equations; b) solve practical problems in one variable
 SOL 8.15  a) solve multistep linear equations in one variable (variable on one and two sides of equations); c) ID properties of operations used to solve
 SOL AII.4  solve (algebraically/graphically) a) absolute value equation/inequalities; b) quadratic equations over complex; c) equations containing rational algebraic expression; d) equations containing radical expressions

Instructional Materials 
 VDOE ESS Lesson: Road Trip  Equations and Inequalities (PDF)  Writing equations to describe reallife data; creating statistical plots (Word)
 VDOE ESS Lesson: Spring Fling Carnival  Equations and Inequalities (PDF)  Writing equations and finding the solutions (intersections) graphically (Word)
 VDOE ESS Lesson: The Exercise Fields  Equations and Inequalities (PDF)  Setting up a system of equations to solve a realworld problem (Word)
 VDOE ESS Lesson: How Much Is That Tune?  Equations and Inequalities, Statistics (PDF)  Solving systems of two linear equations in two variables (Word)
 Notes: Graphing Systems using Substitution (ppt)
 Notes: Graphing Systems using Elimination (ppt)

Real World Lessons 
 Dan Meyer: Playing Catch Up  Who will win?

Practice 
 VDOE ESS: Road Trip Worksheet (pdf)
 VDOE ESS: Salaries Worksheet (pdf)
 VDOE ESS: Saving Money Worksheet (pdf)
 Worksheet: Solving Systems using Substitution CW (doc)
 Worksheet: Solving Systems using Substitution HW (doc)
 Worksheet: Solving Systems using Elimination CW (doc)
 Worksheet: Solving Systems using Elimination HW (doc)
 ExamView Quiz: Solving Systems by Substitution
 Prentice Hall Quiz  Substitution
 ExamView Quiz: Solving Systems by Elimination Using Addition and Subtraction
 Prentice Hall Quiz  Elimination
 ExamView Quiz: Solving Systems by Elimination Using Multiplication

SelfAssessments 

Skills 
 Mathematics Assessment Project:
 A09:Buying Chips and Candy  Apprentice  In this task, you will write equations to solve problems about buying bags of chips and candy bars.
 A12:Printing Tickets  Apprentice  Susie is organizing the printing of tickets for a show. She has collected prices from several printers. Your task is to use graphs and algebra to advise Susie on how to choose the best printer. (This task is very similar to Best Buy Tickets, but offers more stepbystep guidance.)
 A12:Fencing  Apprentice  In this task, you must figure out the cost of building fences from fence posts and wooden panels.
 E10:A Golden Crown?  Expert  Archimedes famously solved a problem for a King who thought his crown might be a fake. In this task, you must work out whether the crown is pure gold.
 E14:Best Buy Tickets  Expert  This task is very similar to A12:Printing Tickets, but offers less stepbystep guidance.
 JMU Pivotal Question: Creating a System of Equations (doc)
 JMU Pivotal Question: Real World System of Equations (doc)

Videos 
 Video: PH Solving Systems Using Substitution
 Video: PH Solving Systems by Substitution to Solve RealWorld Problems
 Video: PH Solving Systems Using Elimination (Adding)
 Video: PH Solving Systems Using Elimination (Multiplying)
 Video Tutor Solving Systems Algebraically

Explore
Learning 
 Modeling Linear Systems  Activity A  Experiment with a system of two lines representing a cat‑and‑mouse chase. Adjust the speeds of the cat and mouse and the head start of the mouse, and immediately see the effects on the graph and on the chase. Connect real‑world meaning to slope, y‑intercept, and the intersection of lines.
 Solving Linear Systems by Graphing  Compare a system of equations in standard form or in slope‑intercept form to its graph. Examine the graph and table of values. Determine the solutions to the system.
 Special Types of Solutions to Linear Systems  Compare a system of equations in standard form to its graph. Examine the graph and table of values. Determine the solutions to the system.
 Systems of Linear Equations  Activity A Solve a system of linear equations by graphing and finding the intersection of the lines of the equations. Create a system of equations, examine its graph, matrix, and table of values, and determine the solution of the system.
 Road Trip (Problem Solving)  Plan a crosscountry road trip through various U.S. state capitals. First choose a vehicle to drive, and then fill up the tank with gas and go! Find the range and gas mileage of each vehicle, and discover the shortest path between two cities.

Websites 
 Interactive
Notes: Solving Systems by Substitution 
lessons and practice problems at Regentsprep.org
 Interactive
Notes: Solving Systems by Elimination  lessons and practice
problems at Regentsprep.org
 Notes: Systems of Equations notes

Old Website Links 

Common Core
Standards 
8.EE.7a 
8.EE.7b 
8.EE.8a 
8.EE.8b 
8.EE.8c 
ASSE.3a 
AAPR.1 
ACED.1 
ACED.2 
ACED.3 
ACED.4 
AREI.1 
AREI.3 
AREI.4a 
AREI.4b 
AREI.5 
AREI.6 
AREI.11 
FIF.8a 

