Grade 8 · Expressions & Equations · MAT-08-EE-002
Solve Equations with Variables on Both Sides
Represent and solve solve equations with variables on both sides problems using precise mathematical notation, multiple representations, and justified reasoning.
Learning objectives
What you should be able to do
- Represent and solve solve equations with variables on both sides problems using precise mathematical notation, multiple representations, and justified reasoning.
- Interpret and verify solve equations with variables on both sides 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
Solve Equations with Variables on Both Sides belongs to expressions & equations. Identify the governing property or relationship before choosing a procedure.
Multiple representations
Represent solve equations with variables on both sides 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 solve equations with variables on both sides 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 2x + 3 = 1x + 5.
1x = 2; x = 2Collect variable terms on one side and constants on the other.
Solve 3x + 5 = 2x + 8.
1x = 3; x = 3Collect variable terms on one side and constants on the other.
Solve 4x + 7 = 3x + 11.
1x = 4; x = 4Collect variable terms on one side and constants on the other.
Solve 5x + 9 = 1x + 29.
4x = 20; x = 5Collect variable terms on one side and constants on the other.
Solve 6x + 3 = 2x + 27.
4x = 24; x = 6Collect variable terms on one side and constants on the other.
Solve 7x + 5 = 3x + 33.
4x = 28; x = 7Collect variable terms on one side and constants on the other.
Solve 8x + 7 = 1x + 63.
7x = 56; x = 8Collect variable terms on one side and constants on the other.
Solve 2x + 9 = 2x + 9.
0x = 0; x = 9Collect variable terms on one side and constants on the other.
Solve 3x + 3 = 3x + 3.
0x = 0; x = 10Collect variable terms on one side and constants on the other.
Solve 4x + 5 = 1x + 11.
3x = 6; x = 2Collect variable terms on one side and constants on the other.
Solve 5x + 7 = 2x + 16.
3x = 9; x = 3Collect variable terms on one side and constants on the other.
Solve 6x + 9 = 3x + 21.
3x = 12; x = 4Collect variable terms on one side and constants on the other.
Solve 7x + 3 = 1x + 33.
6x = 30; x = 5Collect variable terms on one side and constants on the other.
Solve 8x + 5 = 2x + 41.
6x = 36; x = 6Collect variable terms on one side and constants on the other.
Solve 2x + 7 = 3x + 0.
-1x = -7; x = 7Collect variable terms on one side and constants on the other.
Solve 3x + 9 = 1x + 25.
2x = 16; x = 8Collect variable terms on one side and constants on the other.
Solve 4x + 3 = 2x + 21.
2x = 18; x = 9Collect variable terms on one side and constants on the other.
Solve 5x + 5 = 3x + 25.
2x = 20; x = 10Collect variable terms on one side and constants on the other.
Solve 6x + 7 = 1x + 17.
5x = 10; x = 2Collect variable terms on one side and constants on the other.
Solve 7x + 9 = 2x + 24.
5x = 15; x = 3Collect variable terms on one side and constants on the other.
Practice
Guided practice with hints
01Solve 8x + 3 = 3x + 23.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: 5x = 20; x = 4
02Solve 2x + 5 = 1x + 10.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: 1x = 5; x = 5
03Solve 3x + 7 = 2x + 13.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: 1x = 6; x = 6
04Solve 4x + 9 = 3x + 16.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: 1x = 7; x = 7
05Solve 5x + 3 = 1x + 35.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: 4x = 32; x = 8
06Solve 6x + 5 = 2x + 41.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: 4x = 36; x = 9
Practice
Independent practice
01Solve 7x + 7 = 3x + 47.
Answer: 4x = 40; x = 10
02Solve 8x + 9 = 1x + 23.
Answer: 7x = 14; x = 2
03Solve 2x + 3 = 2x + 3.
Answer: 0x = 0; x = 3
04Solve 3x + 5 = 3x + 5.
Answer: 0x = 0; x = 4
05Solve 4x + 7 = 1x + 22.
Answer: 3x = 15; x = 5
06Solve 5x + 9 = 2x + 27.
Answer: 3x = 18; x = 6
07Solve 6x + 3 = 3x + 24.
Answer: 3x = 21; x = 7
08Solve 7x + 5 = 1x + 53.
Answer: 6x = 48; x = 8
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
Solve 8x + 7 = 2x + 61.
6x = 54; x = 9
Solve 2x + 9 = 3x + -1.
-1x = -10; x = 10
Solve 3x + 3 = 1x + 7.
2x = 4; x = 2
Solve 4x + 5 = 2x + 11.
2x = 6; x = 3
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 8 Algebra & Modeling Sprint
Solve solve equations with variables on both sides problems and justify each result with a valid verification method.
Practice
Mastery check
01Solve 5x + 7 = 3x + 15.
Answer: 2x = 8; x = 4
02Solve 6x + 9 = 1x + 34.
Answer: 5x = 25; x = 5
03Solve 7x + 3 = 2x + 33.
Answer: 5x = 30; x = 6
04Solve 8x + 5 = 3x + 40.
Answer: 5x = 35; x = 7
05Solve 2x + 7 = 1x + 15.
Answer: 1x = 8; x = 8
06Solve 3x + 9 = 2x + 18.
Answer: 1x = 9; x = 9
07Solve 4x + 3 = 3x + 13.
Answer: 1x = 10; x = 10
08Solve 5x + 5 = 1x + 13.
Answer: 4x = 8; x = 2
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.