Grade 8 · Expressions & Equations · MAT-08-EE-008
Solve Systems Algebraically
Represent and solve solve systems algebraically problems using precise mathematical notation, multiple representations, and justified reasoning.
Learning objectives
What you should be able to do
- Represent and solve solve systems algebraically problems using precise mathematical notation, multiple representations, and justified reasoning.
- Interpret and verify solve systems algebraically 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 Systems Algebraically belongs to expressions & equations. Identify the governing property or relationship before choosing a procedure.
Multiple representations
Represent solve systems algebraically 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 algebraically 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)Set the expressions for y equal, solve x, then substitute for y.
Solve y=2x+-1 and y=3x+-3.
(2, 3)Set the expressions for y equal, solve x, then substitute for y.
Solve y=3x+-5 and y=4x+-8.
(3, 4)Set the expressions for y equal, solve x, then substitute for y.
Solve y=4x+-11 and y=5x+-15.
(4, 5)Set the expressions for y equal, solve x, then substitute for y.
Solve y=1x+1 and y=2x+-4.
(5, 6)Set the expressions for y equal, solve x, then substitute for y.
Solve y=2x+-5 and y=3x+-11.
(6, 7)Set the expressions for y equal, solve x, then substitute for y.
Solve y=3x+5 and y=4x+4.
(1, 8)Set the expressions for y equal, solve x, then substitute for y.
Solve y=4x+-6 and y=5x+-8.
(2, 2)Set the expressions for y equal, solve x, then substitute for y.
Solve y=1x+0 and y=2x+-3.
(3, 3)Set the expressions for y equal, solve x, then substitute for y.
Solve y=2x+-4 and y=3x+-8.
(4, 4)Set the expressions for y equal, solve x, then substitute for y.
Solve y=3x+-10 and y=4x+-15.
(5, 5)Set the expressions for y equal, solve x, then substitute for y.
Solve y=4x+-18 and y=5x+-24.
(6, 6)Set the expressions for y equal, solve x, then substitute for y.
Solve y=1x+6 and y=2x+5.
(1, 7)Set the expressions for y equal, solve x, then substitute for y.
Solve y=2x+4 and y=3x+2.
(2, 8)Set the expressions for y equal, solve x, then substitute for y.
Solve y=3x+-7 and y=4x+-10.
(3, 2)Set the expressions for y equal, solve x, then substitute for y.
Solve y=4x+-13 and y=5x+-17.
(4, 3)Set the expressions for y equal, solve x, then substitute for y.
Solve y=1x+-1 and y=2x+-6.
(5, 4)Set the expressions for y equal, solve x, then substitute for y.
Solve y=2x+-7 and y=3x+-13.
(6, 5)Set the expressions for y equal, solve x, then substitute for y.
Solve y=3x+3 and y=4x+2.
(1, 6)Set the expressions for y equal, solve x, then substitute for y.
Solve y=4x+-1 and y=5x+-3.
(2, 7)Set the expressions for y equal, solve x, then substitute for y.
Practice
Guided practice with hints
01Solve 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)
02Solve 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)
03Solve y=3x+-12 and y=4x+-17.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: (5, 3)
04Solve y=4x+-20 and y=5x+-26.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: (6, 4)
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)
06Solve 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
01Solve y=3x+-2 and y=4x+-5.
Answer: (3, 7)
02Solve y=4x+-8 and y=5x+-12.
Answer: (4, 8)
03Solve y=1x+-3 and y=2x+-8.
Answer: (5, 2)
04Solve y=2x+-9 and y=3x+-15.
Answer: (6, 3)
05Solve y=3x+1 and y=4x+0.
Answer: (1, 4)
06Solve y=4x+-3 and y=5x+-5.
Answer: (2, 5)
07Solve y=1x+3 and y=2x+0.
Answer: (3, 6)
08Solve 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
Solve y=3x+-7 and y=4x+-12.
(5, 8)
Solve y=4x+-22 and y=5x+-28.
(6, 2)
Solve y=1x+2 and y=2x+1.
(1, 3)
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 algebraically problems and justify each result with a valid verification method.
Practice
Mastery check
01Solve y=3x+-4 and y=4x+-7.
Answer: (3, 5)
02Solve y=4x+-10 and y=5x+-14.
Answer: (4, 6)
03Solve y=1x+2 and y=2x+-3.
Answer: (5, 7)
04Solve y=2x+-4 and y=3x+-10.
Answer: (6, 8)
05Solve y=3x+-1 and y=4x+-2.
Answer: (1, 2)
06Solve y=4x+-5 and y=5x+-7.
Answer: (2, 3)
07Solve y=1x+1 and y=2x+-2.
Answer: (3, 4)
08Solve 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
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.