Chapter 2 · Lesson 7 of 10 · Grade 8 · Expressions & Equations · MAT-08-EE-007
Solve Systems by Graphing
Represent and solve solve systems by graphing problems using precise mathematical notation, multiple representations, and justified reasoning.
Math V4 · Course chapter
Chapter 2: Linear Equations and Systems
Expressions & Equations
Essential question: How can equations and systems represent constraints, intersections, and solution structure?
Where this lesson fits
Lesson 7 of 10. Each lesson builds on earlier chapter representations and ideas rather than resetting the topic.
Chapter destination
Build and defend a multi-representation model using linear equations and systems, including interpretation, verification, and error analysis.
Learning objectives
What you should be able to do
- Represent and solve solve systems by graphing problems using precise mathematical notation, multiple representations, and justified reasoning.
- Interpret and verify solve systems by graphing results in context, including units, constraints, and reasonableness.
Prerequisite check
Make sure the foundation is ready.
Grade 8 uses this prerequisite as active knowledge in a more formal representation.
Learn
Build the idea from meaning, not memorization.
Chapter 2: Linear Equations and Systems
Expressions & Equations. Lesson 7 of 10 extends earlier chapter reasoning.
Connection to the course
Use earlier Grade 8 chapters, Grade 7 prerequisite knowledge as active knowledge. Connect today's representation to at least one earlier representation and explain the relationship.
Meaning and structure
Solve Systems by Graphing belongs to expressions & equations. Identify the governing property or relationship before choosing a procedure.
Multiple representations
Represent solve systems by graphing numerically and symbolically, then connect it to the Grade 8 Math Box. Tables, graphs, equations, coordinate models, diagrams, and verbal descriptions must agree.
Verification and interpretation
Check with substitution, an inverse operation, an equivalent representation, geometric invariants, estimation, or context. State whether the result is exact or approximate and preserve relevant units and constraints.
Math Box · Interactive lesson
Touch the math. Change it. See what stays true.
Use the interactive model before and after the worked examples. Change the inputs, make a prediction, then use the model to test whether your reasoning holds.
Systems Intersection Lab
- Model solve systems by graphing in the Math Box.
- Change one input and predict which quantities or invariants should change.
- Explain what the visual, equation, and numerical result say about the same relationship.
Worked examples
Twenty different ways to see the concept work.
Solve y=1x+1 and y=2x+0.
(1, 2)The solution is the intersection point.
Represent the situation with a table, graph, diagram, number line, or equation before solving: Solve y=2x+-1 and y=3x+-3.
(2, 3)A representation makes the mathematical structure visible before calculation. The solution is the intersection point.
Explain why the method works, not only what steps to perform: Solve y=3x+-5 and y=4x+-8.
(3, 4)Justification connects each step to a definition, property, or relationship. The solution is the intersection point.
Estimate or predict the result first, then calculate and judge whether the answer is reasonable: Solve y=4x+-11 and y=5x+-15.
(4, 5)An estimate provides an independent reasonableness check. The solution is the intersection point.
Verify the result with substitution, an inverse operation, or a second representation: Solve y=1x+1 and y=2x+-4.
(5, 6)Independent verification should support the result without merely repeating the same procedure. The solution is the intersection point.
Interpret the result in context, including units, direction, rate, or geometric meaning: Solve y=2x+-5 and y=3x+-11.
(6, 7)Interpretation connects a computed value to the mathematical situation. The solution is the intersection point.
Error analysis: a student says, "Changing only one side of an equation." Explain the error, then solve this related task correctly: Solve y=3x+5 and y=4x+4.
Apply equivalent operations to both sides. Correct solution: (1, 8)Error analysis identifies the broken idea and replaces it with valid reasoning. The solution is the intersection point.
Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: Solve y=4x+-6 and y=5x+-8. Case B: Solve y=3x+3 and y=4x+2.
Case A: (2, 2) Case B: (1, 6)Comparison should address structure and representation, not only numerical answers. The solution is the intersection point.
Solve y=1x+0 and y=2x+-3.
(3, 3)The solution is the intersection point.
Represent the situation with a table, graph, diagram, number line, or equation before solving: Solve y=2x+-4 and y=3x+-8.
(4, 4)A representation makes the mathematical structure visible before calculation. The solution is the intersection point.
Explain why the method works, not only what steps to perform: Solve y=3x+-10 and y=4x+-15.
(5, 5)Justification connects each step to a definition, property, or relationship. The solution is the intersection point.
Estimate or predict the result first, then calculate and judge whether the answer is reasonable: Solve y=4x+-18 and y=5x+-24.
(6, 6)An estimate provides an independent reasonableness check. The solution is the intersection point.
Verify the result with substitution, an inverse operation, or a second representation: Solve y=1x+6 and y=2x+5.
(1, 7)Independent verification should support the result without merely repeating the same procedure. The solution is the intersection point.
Interpret the result in context, including units, direction, rate, or geometric meaning: Solve y=2x+4 and y=3x+2.
(2, 8)Interpretation connects a computed value to the mathematical situation. The solution is the intersection point.
Error analysis: a student says, "Changing only one side of an equation." Explain the error, then solve this related task correctly: Solve y=3x+-7 and y=4x+-10.
Apply equivalent operations to both sides. Correct solution: (3, 2)Error analysis identifies the broken idea and replaces it with valid reasoning. The solution is the intersection point.
Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: Solve y=4x+-13 and y=5x+-17. Case B: Solve y=3x+-2 and y=4x+-5.
Case A: (4, 3) Case B: (3, 7)Comparison should address structure and representation, not only numerical answers. The solution is the intersection point.
Solve y=1x+-1 and y=2x+-6.
(5, 4)The solution is the intersection point.
Represent the situation with a table, graph, diagram, number line, or equation before solving: Solve y=2x+-7 and y=3x+-13.
(6, 5)A representation makes the mathematical structure visible before calculation. The solution is the intersection point.
Explain why the method works, not only what steps to perform: Solve y=3x+3 and y=4x+2.
(1, 6)Justification connects each step to a definition, property, or relationship. The solution is the intersection point.
Estimate or predict the result first, then calculate and judge whether the answer is reasonable: Solve y=4x+-1 and y=5x+-3.
(2, 7)An estimate provides an independent reasonableness check. The solution is the intersection point.
Practice
Guided practice with hints
01Verify the result with substitution, an inverse operation, or a second representation: Solve y=1x+5 and y=2x+2.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: (3, 8)
02Interpret the result in context, including units, direction, rate, or geometric meaning: Solve y=2x+-6 and y=3x+-10.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: (4, 2)
03Error analysis: a student says, "Changing only one side of an equation." Explain the error, then solve this related task correctly: Solve y=3x+-12 and y=4x+-17.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: Apply equivalent operations to both sides. Correct solution: (5, 3)
04Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: Solve y=4x+-20 and y=5x+-26. Case B: Solve y=3x+-7 and y=4x+-12.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: Case A: (6, 4) Case B: (5, 8)
05Solve y=1x+4 and y=2x+3.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: (1, 5)
06Represent the situation with a table, graph, diagram, number line, or equation before solving: Solve y=2x+2 and y=3x+0.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: (2, 6)
Practice
Independent practice
01Explain why the method works, not only what steps to perform: Solve y=3x+-2 and y=4x+-5.
Answer: (3, 7)
02Estimate or predict the result first, then calculate and judge whether the answer is reasonable: Solve y=4x+-8 and y=5x+-12.
Answer: (4, 8)
03Verify the result with substitution, an inverse operation, or a second representation: Solve y=1x+-3 and y=2x+-8.
Answer: (5, 2)
04Interpret the result in context, including units, direction, rate, or geometric meaning: Solve y=2x+-9 and y=3x+-15.
Answer: (6, 3)
05Error analysis: a student says, "Changing only one side of an equation." Explain the error, then solve this related task correctly: Solve y=3x+1 and y=4x+0.
Answer: Apply equivalent operations to both sides. Correct solution: (1, 4)
06Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: Solve y=4x+-3 and y=5x+-5. Case B: Solve y=3x+-1 and y=4x+-2.
Answer: Case A: (2, 5) Case B: (1, 2)
07Solve y=1x+3 and y=2x+0.
Answer: (3, 6)
08Represent the situation with a table, graph, diagram, number line, or equation before solving: Solve y=2x+-1 and y=3x+-5.
Answer: (4, 7)
Common mistakes
Learn to catch the error, not just the answer.
Apply equivalent operations to both sides.
Systems can have one, zero, or infinitely many solutions.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Model pricing, mixtures, motion, and paired constraints with equations and systems.
- Verify solutions in every original relationship.
Challenge problems
Explain why the method works, not only what steps to perform: Solve y=3x+-7 and y=4x+-12.
(5, 8)
Estimate or predict the result first, then calculate and judge whether the answer is reasonable: Solve y=4x+-22 and y=5x+-28.
(6, 2)
Verify the result with substitution, an inverse operation, or a second representation: Solve y=1x+2 and y=2x+1.
(1, 3)
Interpret the result in context, including units, direction, rate, or geometric meaning: Solve y=2x+0 and y=3x+-2.
(2, 4)
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 8 Algebra & Modeling Sprint
Solve solve systems by graphing problems and justify each result with a valid verification method.
Practice
Mastery check
01Error analysis: a student says, "Changing only one side of an equation." Explain the error, then solve this related task correctly: Solve y=3x+-4 and y=4x+-7.
Answer: Apply equivalent operations to both sides. Correct solution: (3, 5)
02Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: Solve y=4x+-10 and y=5x+-14. Case B: Solve y=3x+-6 and y=4x+-9.
Answer: Case A: (4, 6) Case B: (3, 3)
03Solve y=1x+2 and y=2x+-3.
Answer: (5, 7)
04Represent the situation with a table, graph, diagram, number line, or equation before solving: Solve y=2x+-4 and y=3x+-10.
Answer: (6, 8)
05Explain why the method works, not only what steps to perform: Solve y=3x+-1 and y=4x+-2.
Answer: (1, 2)
06Estimate or predict the result first, then calculate and judge whether the answer is reasonable: Solve y=4x+-5 and y=5x+-7.
Answer: (2, 3)
07Verify the result with substitution, an inverse operation, or a second representation: Solve y=1x+1 and y=2x+-2.
Answer: (3, 4)
08Interpret the result in context, including units, direction, rate, or geometric meaning: Solve y=2x+-3 and y=3x+-7.
Answer: (4, 5)
Terminology
Words to know
- linear equation
- An equation whose variable terms are first degree.
- system
- Two or more equations considered simultaneously.
- solution
- A value or ordered pair making all required equations true.
- equivalent equation
- An equation with the same solution set as another.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
Grade 8 scope, sequence, and public task structure
Coherent sequencing, representations, proof-oriented reasoning, and tasks.Eureka Math² Grade 8 program structure
Module coherence, cumulative practice, and algebraic fluency.enVision Mathematics Grade 8 instructional model
Problem-based entry points, visual learning, modeling, and application.Common Core State Standards for Mathematics — Grade 8
Grade-level content and mathematical-practice expectations.