Grade 8 · Expressions & Equations · MAT-08-EE-005
Model Real Situations with Linear Equations
Represent and solve model real situations with linear equations problems using precise mathematical notation, multiple representations, and justified reasoning.
Learning objectives
What you should be able to do
- Represent and solve model real situations with linear equations problems using precise mathematical notation, multiple representations, and justified reasoning.
- Interpret and verify model real situations with linear equations 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 Real Situations with Linear Equations belongs to expressions & equations. Identify the governing property or relationship before choosing a procedure.
Multiple representations
Represent model real situations with linear equations 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.
Equation Balance
Adjust both sides. Equality means both expressions have the same value.
Balanced: both sides have equal value.
- Model model real situations with linear equations 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.
A service costs $5 plus $3 per unit. The bill is $17. Find units used.
5 + 3u = 17; u = 4The fixed fee is constant and the per-unit charge multiplies the unknown.
A service costs $6 plus $4 per unit. The bill is $26. Find units used.
6 + 4u = 26; u = 5The fixed fee is constant and the per-unit charge multiplies the unknown.
A service costs $7 plus $5 per unit. The bill is $37. Find units used.
7 + 5u = 37; u = 6The fixed fee is constant and the per-unit charge multiplies the unknown.
A service costs $8 plus $6 per unit. The bill is $50. Find units used.
8 + 6u = 50; u = 7The fixed fee is constant and the per-unit charge multiplies the unknown.
A service costs $9 plus $7 per unit. The bill is $65. Find units used.
9 + 7u = 65; u = 8The fixed fee is constant and the per-unit charge multiplies the unknown.
A service costs $10 plus $8 per unit. The bill is $82. Find units used.
10 + 8u = 82; u = 9The fixed fee is constant and the per-unit charge multiplies the unknown.
A service costs $11 plus $3 per unit. The bill is $41. Find units used.
11 + 3u = 41; u = 10The fixed fee is constant and the per-unit charge multiplies the unknown.
A service costs $12 plus $4 per unit. The bill is $56. Find units used.
12 + 4u = 56; u = 11The fixed fee is constant and the per-unit charge multiplies the unknown.
A service costs $5 plus $5 per unit. The bill is $65. Find units used.
5 + 5u = 65; u = 12The fixed fee is constant and the per-unit charge multiplies the unknown.
A service costs $6 plus $6 per unit. The bill is $30. Find units used.
6 + 6u = 30; u = 4The fixed fee is constant and the per-unit charge multiplies the unknown.
A service costs $7 plus $7 per unit. The bill is $42. Find units used.
7 + 7u = 42; u = 5The fixed fee is constant and the per-unit charge multiplies the unknown.
A service costs $8 plus $8 per unit. The bill is $56. Find units used.
8 + 8u = 56; u = 6The fixed fee is constant and the per-unit charge multiplies the unknown.
A service costs $9 plus $3 per unit. The bill is $30. Find units used.
9 + 3u = 30; u = 7The fixed fee is constant and the per-unit charge multiplies the unknown.
A service costs $10 plus $4 per unit. The bill is $42. Find units used.
10 + 4u = 42; u = 8The fixed fee is constant and the per-unit charge multiplies the unknown.
A service costs $11 plus $5 per unit. The bill is $56. Find units used.
11 + 5u = 56; u = 9The fixed fee is constant and the per-unit charge multiplies the unknown.
A service costs $12 plus $6 per unit. The bill is $72. Find units used.
12 + 6u = 72; u = 10The fixed fee is constant and the per-unit charge multiplies the unknown.
A service costs $5 plus $7 per unit. The bill is $82. Find units used.
5 + 7u = 82; u = 11The fixed fee is constant and the per-unit charge multiplies the unknown.
A service costs $6 plus $8 per unit. The bill is $102. Find units used.
6 + 8u = 102; u = 12The fixed fee is constant and the per-unit charge multiplies the unknown.
A service costs $7 plus $3 per unit. The bill is $19. Find units used.
7 + 3u = 19; u = 4The fixed fee is constant and the per-unit charge multiplies the unknown.
A service costs $8 plus $4 per unit. The bill is $28. Find units used.
8 + 4u = 28; u = 5The fixed fee is constant and the per-unit charge multiplies the unknown.
Practice
Guided practice with hints
01A service costs $9 plus $5 per unit. The bill is $39. Find units used.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: 9 + 5u = 39; u = 6
02A service costs $10 plus $6 per unit. The bill is $52. Find units used.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: 10 + 6u = 52; u = 7
03A service costs $11 plus $7 per unit. The bill is $67. Find units used.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: 11 + 7u = 67; u = 8
04A service costs $12 plus $8 per unit. The bill is $84. Find units used.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: 12 + 8u = 84; u = 9
05A service costs $5 plus $3 per unit. The bill is $35. Find units used.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: 5 + 3u = 35; u = 10
06A service costs $6 plus $4 per unit. The bill is $50. Find units used.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: 6 + 4u = 50; u = 11
Practice
Independent practice
01A service costs $7 plus $5 per unit. The bill is $67. Find units used.
Answer: 7 + 5u = 67; u = 12
02A service costs $8 plus $6 per unit. The bill is $32. Find units used.
Answer: 8 + 6u = 32; u = 4
03A service costs $9 plus $7 per unit. The bill is $44. Find units used.
Answer: 9 + 7u = 44; u = 5
04A service costs $10 plus $8 per unit. The bill is $58. Find units used.
Answer: 10 + 8u = 58; u = 6
05A service costs $11 plus $3 per unit. The bill is $32. Find units used.
Answer: 11 + 3u = 32; u = 7
06A service costs $12 plus $4 per unit. The bill is $44. Find units used.
Answer: 12 + 4u = 44; u = 8
07A service costs $5 plus $5 per unit. The bill is $50. Find units used.
Answer: 5 + 5u = 50; u = 9
08A service costs $6 plus $6 per unit. The bill is $66. Find units used.
Answer: 6 + 6u = 66; u = 10
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
A service costs $7 plus $7 per unit. The bill is $84. Find units used.
7 + 7u = 84; u = 11
A service costs $8 plus $8 per unit. The bill is $104. Find units used.
8 + 8u = 104; u = 12
A service costs $9 plus $3 per unit. The bill is $21. Find units used.
9 + 3u = 21; u = 4
A service costs $10 plus $4 per unit. The bill is $30. Find units used.
10 + 4u = 30; u = 5
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 8 Algebra & Modeling Sprint
Solve model real situations with linear equations problems and justify each result with a valid verification method.
Practice
Mastery check
01A service costs $11 plus $5 per unit. The bill is $41. Find units used.
Answer: 11 + 5u = 41; u = 6
02A service costs $12 plus $6 per unit. The bill is $54. Find units used.
Answer: 12 + 6u = 54; u = 7
03A service costs $5 plus $7 per unit. The bill is $61. Find units used.
Answer: 5 + 7u = 61; u = 8
04A service costs $6 plus $8 per unit. The bill is $78. Find units used.
Answer: 6 + 8u = 78; u = 9
05A service costs $7 plus $3 per unit. The bill is $37. Find units used.
Answer: 7 + 3u = 37; u = 10
06A service costs $8 plus $4 per unit. The bill is $52. Find units used.
Answer: 8 + 4u = 52; u = 11
07A service costs $9 plus $5 per unit. The bill is $69. Find units used.
Answer: 9 + 5u = 69; u = 12
08A service costs $10 plus $6 per unit. The bill is $34. Find units used.
Answer: 10 + 6u = 34; u = 4
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.