Grade 7 · Expressions & Equations · MAT-07-EE-001
Simplify Expressions with Rational Coefficients
Represent and solve simplify expressions with rational coefficients problems using precise mathematical notation, models, and reasoning.
Learning objectives
What you should be able to do
- Represent and solve simplify expressions with rational coefficients problems using precise mathematical notation, models, and reasoning.
- Explain why a method for simplify expressions with rational coefficients works, interpret the result in context, and verify it independently.
Prerequisite check
Make sure the foundation is ready.
The new lesson depends on this prerequisite as active knowledge.
The new lesson depends on this prerequisite as active knowledge.
Learn
Build the idea from meaning, not memorization.
Mathematical meaning
Simplify Expressions with Rational Coefficients uses symbols to describe relationships. Equality and inequality are relationships that must be preserved on both sides.
Represent the relationship
The Math Box uses algebra tiles to expose the structure. Change one input, predict the result, then connect the visual change to an equation, table, graph, number line, or geometric model.
Calculate, interpret, and verify
Verify algebra by substituting the proposed solution into the original statement; equivalent expressions must agree for the same input.
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.
Algebra Tiles Lab
Model a linear expression with x-tiles and unit tiles, then evaluate it.
- Model one example of simplify expressions with rational coefficients in the Math Box.
- Change one input, predict the effect, and test the prediction.
- State the relationship or invariant that explains what stayed mathematically consistent.
Worked examples
Twenty different ways to see the concept work.
Combine like terms: -2x + 3x.
(-2 + 3)x = 1xLike terms share the same variable factor.
Combine like terms: 3x + 5x.
(3 + 5)x = 8xLike terms share the same variable factor.
Combine like terms: -4x + 7x.
(-4 + 7)x = 3xLike terms share the same variable factor.
Combine like terms: 5x + 9x.
(5 + 9)x = 14xLike terms share the same variable factor.
Combine like terms: -6x + 3x.
(-6 + 3)x = -3xLike terms share the same variable factor.
Combine like terms: 7x + 5x.
(7 + 5)x = 12xLike terms share the same variable factor.
Combine like terms: -8x + 7x.
(-8 + 7)x = -1xLike terms share the same variable factor.
Combine like terms: 2x + 9x.
(2 + 9)x = 11xLike terms share the same variable factor.
Combine like terms: -3x + 3x.
(-3 + 3)x = 0xLike terms share the same variable factor.
Combine like terms: 4x + 5x.
(4 + 5)x = 9xLike terms share the same variable factor.
Combine like terms: -5x + 7x.
(-5 + 7)x = 2xLike terms share the same variable factor.
Combine like terms: 6x + 9x.
(6 + 9)x = 15xLike terms share the same variable factor.
Combine like terms: -7x + 3x.
(-7 + 3)x = -4xLike terms share the same variable factor.
Combine like terms: 8x + 5x.
(8 + 5)x = 13xLike terms share the same variable factor.
Combine like terms: -2x + 7x.
(-2 + 7)x = 5xLike terms share the same variable factor.
Combine like terms: 3x + 9x.
(3 + 9)x = 12xLike terms share the same variable factor.
Combine like terms: -4x + 3x.
(-4 + 3)x = -1xLike terms share the same variable factor.
Combine like terms: 5x + 5x.
(5 + 5)x = 10xLike terms share the same variable factor.
Combine like terms: -6x + 7x.
(-6 + 7)x = 1xLike terms share the same variable factor.
Combine like terms: 7x + 9x.
(7 + 9)x = 16xLike terms share the same variable factor.
Practice
Guided practice with hints
01Combine like terms: -8x + 3x.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: (-8 + 3)x = -5x
02Combine like terms: 2x + 5x.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: (2 + 5)x = 7x
03Combine like terms: -3x + 7x.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: (-3 + 7)x = 4x
04Combine like terms: 4x + 9x.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: (4 + 9)x = 13x
05Combine like terms: -5x + 3x.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: (-5 + 3)x = -2x
06Combine like terms: 6x + 5x.
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: (6 + 5)x = 11x
Practice
Independent practice
01Combine like terms: -7x + 7x.
Answer: (-7 + 7)x = 0x
02Combine like terms: 8x + 9x.
Answer: (8 + 9)x = 17x
03Combine like terms: -2x + 3x.
Answer: (-2 + 3)x = 1x
04Combine like terms: 3x + 5x.
Answer: (3 + 5)x = 8x
05Combine like terms: -4x + 7x.
Answer: (-4 + 7)x = 3x
06Combine like terms: 5x + 9x.
Answer: (5 + 9)x = 14x
07Combine like terms: -6x + 3x.
Answer: (-6 + 3)x = -3x
08Combine like terms: 7x + 5x.
Answer: (7 + 5)x = 12x
Common mistakes
Learn to catch the error, not just the answer.
Use equivalent operations on both sides.
Only terms with the same variable part are like terms.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Translate cost, distance, rate, and measurement situations into symbolic rules.
- Use equations and inequalities to determine unknown quantities, thresholds, and feasible ranges.
Challenge problems
Combine like terms: -8x + 7x.
(-8 + 7)x = -1x
Combine like terms: 2x + 9x.
(2 + 9)x = 11x
Combine like terms: -3x + 3x.
(-3 + 3)x = 0x
Combine like terms: 4x + 5x.
(4 + 5)x = 9x
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 7 Reasoning Sprint
Solve simplify expressions with rational coefficients problems accurately and justify each result with a valid check.
Practice
Mastery check
01Combine like terms: -5x + 7x.
Answer: (-5 + 7)x = 2x
02Combine like terms: 6x + 9x.
Answer: (6 + 9)x = 15x
03Combine like terms: -7x + 3x.
Answer: (-7 + 3)x = -4x
04Combine like terms: 8x + 5x.
Answer: (8 + 5)x = 13x
05Combine like terms: -2x + 7x.
Answer: (-2 + 7)x = 5x
06Combine like terms: 3x + 9x.
Answer: (3 + 9)x = 12x
07Combine like terms: -4x + 3x.
Answer: (-4 + 3)x = -1x
08Combine like terms: 5x + 5x.
Answer: (5 + 5)x = 10x
Terminology
Words to know
- variable
- A symbol representing a number that may vary or be unknown.
- coefficient
- A numerical factor multiplying a variable.
- expression
- A combination of numbers, variables, and operations.
- solution
- A value that makes an equation or inequality true.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
California Mathematics Framework (2023)
Middle-school instructional guidance, mathematical practices, modeling, and coherent progression.Common Core State Standards for Mathematics — Grade 7
Grade-level expectations for ratios, number systems, algebra, geometry, statistics, and probability.Principles to Actions: Ensuring Mathematical Success for All
Research-informed emphasis on reasoning, representations, discourse, productive struggle, and conceptual understanding.