Grade 8 · Expressions & Equations · MAT-08-EE-010
Model and Verify Systems in Context
Represent and solve model and verify systems in context problems using precise mathematical notation, multiple representations, and justified reasoning.
Learning objectives
What you should be able to do
- Represent and solve model and verify systems in context problems using precise mathematical notation, multiple representations, and justified reasoning.
- Interpret and verify model and verify systems in context 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.
Meaning and structure
Model and Verify Systems in Context belongs to expressions & equations. Identify the governing property or relationship before choosing a procedure.
Multiple representations
Represent model and verify systems in context 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 model and verify systems in context 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.
Adult tickets cost $8 and child tickets $4. 16 tickets bring $104. Find each count.
a+c=16; 8a+4c=104; a=10, c=6Both the count and revenue equations must be satisfied.
Adult tickets cost $9 and child tickets $5. 18 tickets bring $134. Find each count.
a+c=18; 9a+5c=134; a=11, c=7Both the count and revenue equations must be satisfied.
Adult tickets cost $10 and child tickets $6. 20 tickets bring $168. Find each count.
a+c=20; 10a+6c=168; a=12, c=8Both the count and revenue equations must be satisfied.
Adult tickets cost $11 and child tickets $7. 22 tickets bring $206. Find each count.
a+c=22; 11a+7c=206; a=13, c=9Both the count and revenue equations must be satisfied.
Adult tickets cost $12 and child tickets $4. 24 tickets bring $208. Find each count.
a+c=24; 12a+4c=208; a=14, c=10Both the count and revenue equations must be satisfied.
Adult tickets cost $8 and child tickets $5. 21 tickets bring $150. Find each count.
a+c=21; 8a+5c=150; a=15, c=6Both the count and revenue equations must be satisfied.
Adult tickets cost $9 and child tickets $6. 17 tickets bring $132. Find each count.
a+c=17; 9a+6c=132; a=10, c=7Both the count and revenue equations must be satisfied.
Adult tickets cost $10 and child tickets $7. 19 tickets bring $166. Find each count.
a+c=19; 10a+7c=166; a=11, c=8Both the count and revenue equations must be satisfied.
Adult tickets cost $11 and child tickets $4. 21 tickets bring $168. Find each count.
a+c=21; 11a+4c=168; a=12, c=9Both the count and revenue equations must be satisfied.
Adult tickets cost $12 and child tickets $5. 23 tickets bring $206. Find each count.
a+c=23; 12a+5c=206; a=13, c=10Both the count and revenue equations must be satisfied.
Adult tickets cost $8 and child tickets $6. 20 tickets bring $148. Find each count.
a+c=20; 8a+6c=148; a=14, c=6Both the count and revenue equations must be satisfied.
Adult tickets cost $9 and child tickets $7. 22 tickets bring $184. Find each count.
a+c=22; 9a+7c=184; a=15, c=7Both the count and revenue equations must be satisfied.
Adult tickets cost $10 and child tickets $4. 18 tickets bring $132. Find each count.
a+c=18; 10a+4c=132; a=10, c=8Both the count and revenue equations must be satisfied.
Adult tickets cost $11 and child tickets $5. 20 tickets bring $166. Find each count.
a+c=20; 11a+5c=166; a=11, c=9Both the count and revenue equations must be satisfied.
Adult tickets cost $12 and child tickets $6. 22 tickets bring $204. Find each count.
a+c=22; 12a+6c=204; a=12, c=10Both the count and revenue equations must be satisfied.
Adult tickets cost $8 and child tickets $7. 19 tickets bring $146. Find each count.
a+c=19; 8a+7c=146; a=13, c=6Both the count and revenue equations must be satisfied.
Adult tickets cost $9 and child tickets $4. 21 tickets bring $154. Find each count.
a+c=21; 9a+4c=154; a=14, c=7Both the count and revenue equations must be satisfied.
Adult tickets cost $10 and child tickets $5. 23 tickets bring $190. Find each count.
a+c=23; 10a+5c=190; a=15, c=8Both the count and revenue equations must be satisfied.
Adult tickets cost $11 and child tickets $6. 19 tickets bring $164. Find each count.
a+c=19; 11a+6c=164; a=10, c=9Both the count and revenue equations must be satisfied.
Adult tickets cost $12 and child tickets $7. 21 tickets bring $202. Find each count.
a+c=21; 12a+7c=202; a=11, c=10Both the count and revenue equations must be satisfied.
Practice
Guided practice with hints
01Adult tickets cost $8 and child tickets $4. 18 tickets bring $120. Find each count.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: a+c=18; 8a+4c=120; a=12, c=6
02Adult tickets cost $9 and child tickets $5. 20 tickets bring $152. Find each count.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: a+c=20; 9a+5c=152; a=13, c=7
03Adult tickets cost $10 and child tickets $6. 22 tickets bring $188. Find each count.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: a+c=22; 10a+6c=188; a=14, c=8
04Adult tickets cost $11 and child tickets $7. 24 tickets bring $228. Find each count.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: a+c=24; 11a+7c=228; a=15, c=9
05Adult tickets cost $12 and child tickets $4. 20 tickets bring $160. Find each count.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: a+c=20; 12a+4c=160; a=10, c=10
06Adult tickets cost $8 and child tickets $5. 17 tickets bring $118. Find each count.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: a+c=17; 8a+5c=118; a=11, c=6
Practice
Independent practice
01Adult tickets cost $9 and child tickets $6. 19 tickets bring $150. Find each count.
Answer: a+c=19; 9a+6c=150; a=12, c=7
02Adult tickets cost $10 and child tickets $7. 21 tickets bring $186. Find each count.
Answer: a+c=21; 10a+7c=186; a=13, c=8
03Adult tickets cost $11 and child tickets $4. 23 tickets bring $190. Find each count.
Answer: a+c=23; 11a+4c=190; a=14, c=9
04Adult tickets cost $12 and child tickets $5. 25 tickets bring $230. Find each count.
Answer: a+c=25; 12a+5c=230; a=15, c=10
05Adult tickets cost $8 and child tickets $6. 16 tickets bring $116. Find each count.
Answer: a+c=16; 8a+6c=116; a=10, c=6
06Adult tickets cost $9 and child tickets $7. 18 tickets bring $148. Find each count.
Answer: a+c=18; 9a+7c=148; a=11, c=7
07Adult tickets cost $10 and child tickets $4. 20 tickets bring $152. Find each count.
Answer: a+c=20; 10a+4c=152; a=12, c=8
08Adult tickets cost $11 and child tickets $5. 22 tickets bring $188. Find each count.
Answer: a+c=22; 11a+5c=188; a=13, c=9
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
Adult tickets cost $12 and child tickets $6. 24 tickets bring $228. Find each count.
a+c=24; 12a+6c=228; a=14, c=10
Adult tickets cost $8 and child tickets $7. 21 tickets bring $162. Find each count.
a+c=21; 8a+7c=162; a=15, c=6
Adult tickets cost $9 and child tickets $4. 17 tickets bring $118. Find each count.
a+c=17; 9a+4c=118; a=10, c=7
Adult tickets cost $10 and child tickets $5. 19 tickets bring $150. Find each count.
a+c=19; 10a+5c=150; a=11, c=8
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 8 Algebra & Modeling Sprint
Solve model and verify systems in context problems and justify each result with a valid verification method.
Practice
Mastery check
01Adult tickets cost $11 and child tickets $6. 21 tickets bring $186. Find each count.
Answer: a+c=21; 11a+6c=186; a=12, c=9
02Adult tickets cost $12 and child tickets $7. 23 tickets bring $226. Find each count.
Answer: a+c=23; 12a+7c=226; a=13, c=10
03Adult tickets cost $8 and child tickets $4. 20 tickets bring $136. Find each count.
Answer: a+c=20; 8a+4c=136; a=14, c=6
04Adult tickets cost $9 and child tickets $5. 22 tickets bring $170. Find each count.
Answer: a+c=22; 9a+5c=170; a=15, c=7
05Adult tickets cost $10 and child tickets $6. 18 tickets bring $148. Find each count.
Answer: a+c=18; 10a+6c=148; a=10, c=8
06Adult tickets cost $11 and child tickets $7. 20 tickets bring $184. Find each count.
Answer: a+c=20; 11a+7c=184; a=11, c=9
07Adult tickets cost $12 and child tickets $4. 22 tickets bring $184. Find each count.
Answer: a+c=22; 12a+4c=184; a=12, c=10
08Adult tickets cost $8 and child tickets $5. 19 tickets bring $134. Find each count.
Answer: a+c=19; 8a+5c=134; a=13, c=6
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
California Mathematics Framework (2023)
Grade 8 instructional guidance, mathematical practices, modeling, and progression toward high-school mathematics.Common Core State Standards for Mathematics — Grade 8
Grade-level expectations for real numbers, equations, functions, geometry, and statistics.Principles to Actions: Ensuring Mathematical Success for All
Research-informed emphasis on reasoning, representations, discourse, productive struggle, and conceptual understanding.