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Math V4 · Course chapter

Chapter 4: Expressions, Equations, and Inequalities

Using symbols to describe relationships

Essential question: How can a verbal or numerical relationship be expressed symbolically, transformed without changing its meaning, and solved?

Where this lesson fits

Lesson 6 of 10. Each lesson builds on earlier chapter representations and ideas rather than resetting the topic.

Chapter destination

Model a real pricing or distance situation with variables, expressions, equations, inequalities, tables, and graphs.

Learning objectives

What you should be able to do

Represent and solve understand solutions to equations problems using precise mathematical notation, models, and reasoning.
Explain why a method for understand solutions to equations works, interpret the result in context, and verify it independently.

Prerequisite check

Make sure the foundation is ready.

Can you explain and use prerequisite MAT-06-EE-002 without relying on a keyword?

The new lesson depends on this prerequisite as active knowledge.

Learn

Build the idea from meaning, not memorization.

Chapter 4: Expressions, Equations, and Inequalities

Using symbols to describe relationships. This is Lesson 6 of 10 in Chapter 4. It builds on earlier chapter ideas instead of restarting the topic from scratch.

Connection to the course

Cumulative knowledge used here includes Chapter 1 rates, Chapter 2 operation fluency, Chapter 3 signed quantities. As you work, connect today's idea to earlier lessons in the chapter and keep those earlier representations available for comparison.

Mathematical meaning

Understand Solutions to Equations uses symbols to describe relationships. Equality and inequality are relationships that must be preserved on both sides.

Represent the relationship

The Math Box uses equation balance 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.

Equation Balance

Adjust both sides. Equality means both expressions have the same value.

Left side8Right side8

8=8

Balanced: both sides have equal value.

Try these actions

Model one example of understand solutions to equations in the Math Box.
Change one input, predict the effect, and test the prediction.
Connect the model to another representation used earlier in this chapter.
State the relationship or invariant that explains what stayed mathematically consistent.

Worked examples

Twenty different ways to see the concept work.

Check whether x = 2 solves 2x + 3 = 7.

2(2) + 3 = 7; yes

A solution makes the equation true when substituted.

Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Check whether x = 3 solves 3x + 5 = 14.

3(3) + 5 = 14; yes

The representation should show the same mathematical relationship as the calculation. A solution makes the equation true when substituted.

Explain why a valid method works, then solve: Check whether x = 4 solves 4x + 7 = 23.

4(4) + 7 = 23; yes

A complete explanation names the relationship or property being preserved. A solution makes the equation true when substituted.

Estimate or predict first, then calculate and decide whether the result is reasonable: Check whether x = 5 solves 5x + 9 = 34.

5(5) + 9 = 34; yes

The estimate is a reasonableness check, not a replacement for the exact result. A solution makes the equation true when substituted.

Solve and verify the result with a second method, inverse operation, or equivalent representation: Check whether x = 6 solves 6x + 3 = 39.

6(6) + 3 = 39; yes

Verification should independently support the result. A solution makes the equation true when substituted.

Interpret the answer in context after solving. What does the result mean here? Check whether x = 7 solves 7x + 5 = 54.

7(7) + 5 = 54; yes

State the result with its meaning, units, direction, or comparison—not only a number. A solution makes the equation true when substituted.

Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Check whether x = 8 solves 8x + 7 = 71.

Use equivalent operations on both sides. Correct solution: 8(8) + 7 = 71; yes

Error analysis requires identifying the broken idea, not merely replacing the final answer. A solution makes the equation true when substituted.

Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Check whether x = 9 solves 2x + 9 = 27. Case B: Check whether x = 9 solves 4x + 3 = 39.

Case A: 2(9) + 9 = 27; yes Case B: 4(9) + 3 = 39; yes

Compare the structure, representation, units, and result rather than only the numbers. A solution makes the equation true when substituted.

Check whether x = 10 solves 3x + 3 = 33.

3(10) + 3 = 33; yes

A solution makes the equation true when substituted.

Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Check whether x = 2 solves 4x + 5 = 13.

4(2) + 5 = 13; yes

The representation should show the same mathematical relationship as the calculation. A solution makes the equation true when substituted.

Explain why a valid method works, then solve: Check whether x = 3 solves 5x + 7 = 22.

5(3) + 7 = 22; yes

A complete explanation names the relationship or property being preserved. A solution makes the equation true when substituted.

Estimate or predict first, then calculate and decide whether the result is reasonable: Check whether x = 4 solves 6x + 9 = 33.

6(4) + 9 = 33; yes

The estimate is a reasonableness check, not a replacement for the exact result. A solution makes the equation true when substituted.

Solve and verify the result with a second method, inverse operation, or equivalent representation: Check whether x = 5 solves 7x + 3 = 38.

7(5) + 3 = 38; yes

Verification should independently support the result. A solution makes the equation true when substituted.

Interpret the answer in context after solving. What does the result mean here? Check whether x = 6 solves 8x + 5 = 53.

8(6) + 5 = 53; yes

State the result with its meaning, units, direction, or comparison—not only a number. A solution makes the equation true when substituted.

Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Check whether x = 7 solves 2x + 7 = 21.

Use equivalent operations on both sides. Correct solution: 2(7) + 7 = 21; yes

Error analysis requires identifying the broken idea, not merely replacing the final answer. A solution makes the equation true when substituted.

Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Check whether x = 8 solves 3x + 9 = 33. Case B: Check whether x = 8 solves 5x + 3 = 43.

Case A: 3(8) + 9 = 33; yes Case B: 5(8) + 3 = 43; yes

Compare the structure, representation, units, and result rather than only the numbers. A solution makes the equation true when substituted.

Check whether x = 9 solves 4x + 3 = 39.

4(9) + 3 = 39; yes

A solution makes the equation true when substituted.

Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Check whether x = 10 solves 5x + 5 = 55.

5(10) + 5 = 55; yes

The representation should show the same mathematical relationship as the calculation. A solution makes the equation true when substituted.

Explain why a valid method works, then solve: Check whether x = 2 solves 6x + 7 = 19.

6(2) + 7 = 19; yes

A complete explanation names the relationship or property being preserved. A solution makes the equation true when substituted.

Estimate or predict first, then calculate and decide whether the result is reasonable: Check whether x = 3 solves 7x + 9 = 30.

7(3) + 9 = 30; yes

The estimate is a reasonableness check, not a replacement for the exact result. A solution makes the equation true when substituted.

Practice

Guided practice with hints

01Solve and verify the result with a second method, inverse operation, or equivalent representation: Check whether x = 4 solves 8x + 3 = 35.

Hint: Choose a representation before computing.

Answer: 8(4) + 3 = 35; yes

02Interpret the answer in context after solving. What does the result mean here? Check whether x = 5 solves 2x + 5 = 15.

Hint: Explain what relationship or property makes your method valid.

Answer: 2(5) + 5 = 15; yes

03Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Check whether x = 6 solves 3x + 7 = 25.

Hint: Choose a representation before computing.

Answer: Use equivalent operations on both sides. Correct solution: 3(6) + 7 = 25; yes

04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Check whether x = 7 solves 4x + 9 = 37. Case B: Check whether x = 7 solves 6x + 3 = 45.

Hint: Explain what relationship or property makes your method valid.

Answer: Case A: 4(7) + 9 = 37; yes Case B: 6(7) + 3 = 45; yes

05Check whether x = 8 solves 5x + 3 = 43.

Hint: Choose a representation before computing.

Answer: 5(8) + 3 = 43; yes

06Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Check whether x = 9 solves 6x + 5 = 59.

Hint: Explain what relationship or property makes your method valid.

Answer: 6(9) + 5 = 59; yes

Practice

Independent practice

01Explain why a valid method works, then solve: Check whether x = 10 solves 7x + 7 = 77.

Answer: 7(10) + 7 = 77; yes

02Estimate or predict first, then calculate and decide whether the result is reasonable: Check whether x = 2 solves 8x + 9 = 25.

Answer: 8(2) + 9 = 25; yes

03Solve and verify the result with a second method, inverse operation, or equivalent representation: Check whether x = 3 solves 2x + 3 = 9.

Answer: 2(3) + 3 = 9; yes

04Interpret the answer in context after solving. What does the result mean here? Check whether x = 4 solves 3x + 5 = 17.

Answer: 3(4) + 5 = 17; yes

05Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Check whether x = 5 solves 4x + 7 = 27.

Answer: Use equivalent operations on both sides. Correct solution: 4(5) + 7 = 27; yes

06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Check whether x = 6 solves 5x + 9 = 39. Case B: Check whether x = 6 solves 7x + 3 = 45.

Answer: Case A: 5(6) + 9 = 39; yes Case B: 7(6) + 3 = 45; yes

07Check whether x = 7 solves 6x + 3 = 45.

Answer: 6(7) + 3 = 45; yes

08Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Check whether x = 8 solves 7x + 5 = 61.

Answer: 7(8) + 5 = 61; yes

Common mistakes

Learn to catch the error, not just the answer.

Changing only one side of an equation.

Use equivalent operations on both sides.

Combining unlike terms.

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

Explain why a valid method works, then solve: Check whether x = 9 solves 8x + 7 = 79.

8(9) + 7 = 79; yes

Estimate or predict first, then calculate and decide whether the result is reasonable: Check whether x = 10 solves 2x + 9 = 29.

2(10) + 9 = 29; yes

Solve and verify the result with a second method, inverse operation, or equivalent representation: Check whether x = 2 solves 3x + 3 = 9.

3(2) + 3 = 9; yes

Interpret the answer in context after solving. What does the result mean here? Check whether x = 3 solves 4x + 5 = 17.

4(3) + 5 = 17; yes

Flashcards

Retrieve it from memory.

Flashcards

Card 1 of 4

Game mode

Grade 6 Chapter 4 Reasoning Lab

Solve understand solutions to equations problems accurately, choose an appropriate representation, and justify each result with a check.

Round 1 · Score 0Explain why a valid method works, then solve: Check whether x = 9 solves 8x + 7 = 79.

1 point for an accurate result and 1 point for a mathematically valid explanation, representation, or check.

Practice

Mastery check

01Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Check whether x = 4 solves 5x + 7 = 27.

Answer: Use equivalent operations on both sides. Correct solution: 5(4) + 7 = 27; yes

02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Check whether x = 5 solves 6x + 9 = 39. Case B: Check whether x = 5 solves 8x + 3 = 43.

Answer: Case A: 6(5) + 9 = 39; yes Case B: 8(5) + 3 = 43; yes

03Check whether x = 6 solves 7x + 3 = 45.

Answer: 7(6) + 3 = 45; yes

04Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Check whether x = 7 solves 8x + 5 = 61.

Answer: 8(7) + 5 = 61; yes

05Explain why a valid method works, then solve: Check whether x = 8 solves 2x + 7 = 23.

Answer: 2(8) + 7 = 23; yes

06Estimate or predict first, then calculate and decide whether the result is reasonable: Check whether x = 9 solves 3x + 9 = 36.

Answer: 3(9) + 9 = 36; yes

07Solve and verify the result with a second method, inverse operation, or equivalent representation: Check whether x = 10 solves 4x + 3 = 43.

Answer: 4(10) + 3 = 43; yes

08Interpret the answer in context after solving. What does the result mean here? Check whether x = 2 solves 5x + 5 = 15.

Answer: 5(2) + 5 = 15; yes

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

Illustrative Mathematics

Grade 6 scope, sequence, standards, and public task structure

Reference for coherent sequencing, representations, dependency-aware progression, and reasoning-rich task types.

Great Minds

Eureka Math² Grade 6 program structure

Reference for module/topic coherence, recap, mixed-practice, and cumulative-learning patterns.

Savvas Learning Company

enVision Mathematics Grade 6 instructional model

Reference for problem-based entry points, visual learning, modeling, and application patterns.

Common Core State Standards Initiative

Common Core State Standards for Mathematics — Grade 6

Grade-level content and mathematical-practice expectations.

California Mathematics Framework ↗

Math curriculum Next lesson →

Math V4 · Course chapter

Chapter 4: Expressions, Equations, and Inequalities

Using symbols to describe relationships

Essential question: How can a verbal or numerical relationship be expressed symbolically, transformed without changing its meaning, and solved?

Where this lesson fits

Lesson 6 of 10. Each lesson builds on earlier chapter representations and ideas rather than resetting the topic.

Chapter destination

Model a real pricing or distance situation with variables, expressions, equations, inequalities, tables, and graphs.

Learning objectives

What you should be able to do

Represent and solve understand solutions to equations problems using precise mathematical notation, models, and reasoning.
Explain why a method for understand solutions to equations works, interpret the result in context, and verify it independently.

Prerequisite check

Make sure the foundation is ready.

Can you explain and use prerequisite MAT-06-EE-002 without relying on a keyword?

The new lesson depends on this prerequisite as active knowledge.

Learn

Build the idea from meaning, not memorization.

Chapter 4: Expressions, Equations, and Inequalities

Using symbols to describe relationships. This is Lesson 6 of 10 in Chapter 4. It builds on earlier chapter ideas instead of restarting the topic from scratch.

Connection to the course

Mathematical meaning

Understand Solutions to Equations uses symbols to describe relationships. Equality and inequality are relationships that must be preserved on both sides.

Represent the relationship

The Math Box uses equation balance 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.

Equation Balance

Adjust both sides. Equality means both expressions have the same value.

Left side8Right side8

8=8

Balanced: both sides have equal value.

Try these actions

Model one example of understand solutions to equations in the Math Box.
Change one input, predict the effect, and test the prediction.
Connect the model to another representation used earlier in this chapter.
State the relationship or invariant that explains what stayed mathematically consistent.

Worked examples

Twenty different ways to see the concept work.

Check whether x = 2 solves 2x + 3 = 7.

2(2) + 3 = 7; yes

A solution makes the equation true when substituted.

Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Check whether x = 3 solves 3x + 5 = 14.

3(3) + 5 = 14; yes

The representation should show the same mathematical relationship as the calculation. A solution makes the equation true when substituted.

Explain why a valid method works, then solve: Check whether x = 4 solves 4x + 7 = 23.

4(4) + 7 = 23; yes

A complete explanation names the relationship or property being preserved. A solution makes the equation true when substituted.

Estimate or predict first, then calculate and decide whether the result is reasonable: Check whether x = 5 solves 5x + 9 = 34.

5(5) + 9 = 34; yes

The estimate is a reasonableness check, not a replacement for the exact result. A solution makes the equation true when substituted.

Solve and verify the result with a second method, inverse operation, or equivalent representation: Check whether x = 6 solves 6x + 3 = 39.

6(6) + 3 = 39; yes

Verification should independently support the result. A solution makes the equation true when substituted.

Interpret the answer in context after solving. What does the result mean here? Check whether x = 7 solves 7x + 5 = 54.

7(7) + 5 = 54; yes

State the result with its meaning, units, direction, or comparison—not only a number. A solution makes the equation true when substituted.

Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Check whether x = 8 solves 8x + 7 = 71.

Use equivalent operations on both sides. Correct solution: 8(8) + 7 = 71; yes

Error analysis requires identifying the broken idea, not merely replacing the final answer. A solution makes the equation true when substituted.

Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Check whether x = 9 solves 2x + 9 = 27. Case B: Check whether x = 9 solves 4x + 3 = 39.

Case A: 2(9) + 9 = 27; yes Case B: 4(9) + 3 = 39; yes

Compare the structure, representation, units, and result rather than only the numbers. A solution makes the equation true when substituted.

Check whether x = 10 solves 3x + 3 = 33.

3(10) + 3 = 33; yes

A solution makes the equation true when substituted.

Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Check whether x = 2 solves 4x + 5 = 13.

4(2) + 5 = 13; yes

The representation should show the same mathematical relationship as the calculation. A solution makes the equation true when substituted.

Explain why a valid method works, then solve: Check whether x = 3 solves 5x + 7 = 22.

5(3) + 7 = 22; yes

A complete explanation names the relationship or property being preserved. A solution makes the equation true when substituted.

Estimate or predict first, then calculate and decide whether the result is reasonable: Check whether x = 4 solves 6x + 9 = 33.

6(4) + 9 = 33; yes

The estimate is a reasonableness check, not a replacement for the exact result. A solution makes the equation true when substituted.

Solve and verify the result with a second method, inverse operation, or equivalent representation: Check whether x = 5 solves 7x + 3 = 38.

7(5) + 3 = 38; yes

Verification should independently support the result. A solution makes the equation true when substituted.

Interpret the answer in context after solving. What does the result mean here? Check whether x = 6 solves 8x + 5 = 53.

8(6) + 5 = 53; yes

State the result with its meaning, units, direction, or comparison—not only a number. A solution makes the equation true when substituted.

Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Check whether x = 7 solves 2x + 7 = 21.

Use equivalent operations on both sides. Correct solution: 2(7) + 7 = 21; yes

Error analysis requires identifying the broken idea, not merely replacing the final answer. A solution makes the equation true when substituted.

Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Check whether x = 8 solves 3x + 9 = 33. Case B: Check whether x = 8 solves 5x + 3 = 43.

Case A: 3(8) + 9 = 33; yes Case B: 5(8) + 3 = 43; yes

Compare the structure, representation, units, and result rather than only the numbers. A solution makes the equation true when substituted.

Check whether x = 9 solves 4x + 3 = 39.

4(9) + 3 = 39; yes

A solution makes the equation true when substituted.

Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Check whether x = 10 solves 5x + 5 = 55.

5(10) + 5 = 55; yes

The representation should show the same mathematical relationship as the calculation. A solution makes the equation true when substituted.

Explain why a valid method works, then solve: Check whether x = 2 solves 6x + 7 = 19.

6(2) + 7 = 19; yes

A complete explanation names the relationship or property being preserved. A solution makes the equation true when substituted.

Estimate or predict first, then calculate and decide whether the result is reasonable: Check whether x = 3 solves 7x + 9 = 30.

7(3) + 9 = 30; yes

The estimate is a reasonableness check, not a replacement for the exact result. A solution makes the equation true when substituted.

Practice

Guided practice with hints

01Solve and verify the result with a second method, inverse operation, or equivalent representation: Check whether x = 4 solves 8x + 3 = 35.

Hint: Choose a representation before computing.

Answer: 8(4) + 3 = 35; yes

02Interpret the answer in context after solving. What does the result mean here? Check whether x = 5 solves 2x + 5 = 15.

Hint: Explain what relationship or property makes your method valid.

Answer: 2(5) + 5 = 15; yes

03Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Check whether x = 6 solves 3x + 7 = 25.

Hint: Choose a representation before computing.

Answer: Use equivalent operations on both sides. Correct solution: 3(6) + 7 = 25; yes

04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Check whether x = 7 solves 4x + 9 = 37. Case B: Check whether x = 7 solves 6x + 3 = 45.

Hint: Explain what relationship or property makes your method valid.

Answer: Case A: 4(7) + 9 = 37; yes Case B: 6(7) + 3 = 45; yes

05Check whether x = 8 solves 5x + 3 = 43.

Hint: Choose a representation before computing.

Answer: 5(8) + 3 = 43; yes

06Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Check whether x = 9 solves 6x + 5 = 59.

Hint: Explain what relationship or property makes your method valid.

Answer: 6(9) + 5 = 59; yes

Practice

Independent practice

01Explain why a valid method works, then solve: Check whether x = 10 solves 7x + 7 = 77.

Answer: 7(10) + 7 = 77; yes

02Estimate or predict first, then calculate and decide whether the result is reasonable: Check whether x = 2 solves 8x + 9 = 25.

Answer: 8(2) + 9 = 25; yes

03Solve and verify the result with a second method, inverse operation, or equivalent representation: Check whether x = 3 solves 2x + 3 = 9.

Answer: 2(3) + 3 = 9; yes

04Interpret the answer in context after solving. What does the result mean here? Check whether x = 4 solves 3x + 5 = 17.

Answer: 3(4) + 5 = 17; yes

05Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Check whether x = 5 solves 4x + 7 = 27.

Answer: Use equivalent operations on both sides. Correct solution: 4(5) + 7 = 27; yes

06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Check whether x = 6 solves 5x + 9 = 39. Case B: Check whether x = 6 solves 7x + 3 = 45.

Answer: Case A: 5(6) + 9 = 39; yes Case B: 7(6) + 3 = 45; yes

07Check whether x = 7 solves 6x + 3 = 45.

Answer: 6(7) + 3 = 45; yes

08Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Check whether x = 8 solves 7x + 5 = 61.

Answer: 7(8) + 5 = 61; yes

Common mistakes

Learn to catch the error, not just the answer.

Changing only one side of an equation.

Use equivalent operations on both sides.

Combining unlike terms.

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

Explain why a valid method works, then solve: Check whether x = 9 solves 8x + 7 = 79.

8(9) + 7 = 79; yes

Estimate or predict first, then calculate and decide whether the result is reasonable: Check whether x = 10 solves 2x + 9 = 29.

2(10) + 9 = 29; yes

Solve and verify the result with a second method, inverse operation, or equivalent representation: Check whether x = 2 solves 3x + 3 = 9.

3(2) + 3 = 9; yes

Interpret the answer in context after solving. What does the result mean here? Check whether x = 3 solves 4x + 5 = 17.

4(3) + 5 = 17; yes

Flashcards

Retrieve it from memory.

Flashcards

Card 1 of 4

Game mode

Grade 6 Chapter 4 Reasoning Lab

Solve understand solutions to equations problems accurately, choose an appropriate representation, and justify each result with a check.

Round 1 · Score 0Explain why a valid method works, then solve: Check whether x = 9 solves 8x + 7 = 79.

1 point for an accurate result and 1 point for a mathematically valid explanation, representation, or check.

Practice

Mastery check

01Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Check whether x = 4 solves 5x + 7 = 27.

Answer: Use equivalent operations on both sides. Correct solution: 5(4) + 7 = 27; yes

02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Check whether x = 5 solves 6x + 9 = 39. Case B: Check whether x = 5 solves 8x + 3 = 43.

Answer: Case A: 6(5) + 9 = 39; yes Case B: 8(5) + 3 = 43; yes

03Check whether x = 6 solves 7x + 3 = 45.

Answer: 7(6) + 3 = 45; yes

04Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Check whether x = 7 solves 8x + 5 = 61.

Answer: 8(7) + 5 = 61; yes

05Explain why a valid method works, then solve: Check whether x = 8 solves 2x + 7 = 23.

Answer: 2(8) + 7 = 23; yes

06Estimate or predict first, then calculate and decide whether the result is reasonable: Check whether x = 9 solves 3x + 9 = 36.

Answer: 3(9) + 9 = 36; yes

07Solve and verify the result with a second method, inverse operation, or equivalent representation: Check whether x = 10 solves 4x + 3 = 43.

Answer: 4(10) + 3 = 43; yes

08Interpret the answer in context after solving. What does the result mean here? Check whether x = 2 solves 5x + 5 = 15.

Answer: 5(2) + 5 = 15; yes

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

Illustrative Mathematics

Grade 6 scope, sequence, standards, and public task structure

Reference for coherent sequencing, representations, dependency-aware progression, and reasoning-rich task types.

Great Minds

Eureka Math² Grade 6 program structure

Reference for module/topic coherence, recap, mixed-practice, and cumulative-learning patterns.

Savvas Learning Company

enVision Mathematics Grade 6 instructional model

Reference for problem-based entry points, visual learning, modeling, and application patterns.

Common Core State Standards Initiative

Common Core State Standards for Mathematics — Grade 6

Grade-level content and mathematical-practice expectations.

California Mathematics Framework ↗

Math curriculum Next lesson →