CCSS: CCSS.MATH.CONTENT.HSA.REI.B.3 · Difficulty: medium
Fast Track
22 min
Standard
30 min
Thorough
38 min
Learning Objectives (What You'll Master Today)
Solve linear equations in one variable using inverse operations and properties of equality
Translate real-world scenarios into linear equations and interpret solutions in context
Solve equations with variables on both sides and with fractional or decimal coefficients
Justify each step in the solution process using algebraic properties
I Can... I can solve linear equations in one variable using inverse operations and properties of equality
(A) Mathematical Concepts - Self-Learning Guide
Big Idea: Key Concept
A linear equation in one variable is a mathematical sentence stating that two expressions are equal, where the variable appears only to the first power. Solving the equation means finding the value of the variable that makes the equation true. The strategy is straightforward: use inverse operations to isolate the variable on one side. Whatever operation you perform on one side, you must perform on the other to maintain balance. Think of the equation like a balanced scale — if you add weight to one side, you must add the same weight to the other. Linear equations appear everywhere in daily life: calculating costs, converting temperatures, splitting bills, planning budgets, and mixing solutions. Mastering this skill is the foundation for all of algebra and beyond.
🧪 Interactive Simulation
Properties of Equality:If a = b, then a + c = b + c (Addition Property)If a = b, then a − c = b − c (Subtraction Property)If a = b, then a · c = b · c (Multiplication Property)If a = b and c ≠ 0, then a / c = b / c (Division Property)
1Simplify Both Sides: Distribute any parentheses and combine like terms on each side of the equation separately.
2Collect Variable Terms: Use addition or subtraction to move all variable terms to one side of the equation.
3Collect Constant Terms: Use addition or subtraction to move all constant terms to the opposite side.
How-To Methods
Simplify Both Sides: Distribute any parentheses and combine like terms on each side of the equation separately.
Collect Variable Terms: Use addition or subtraction to move all variable terms to one side of the equation.
Collect Constant Terms: Use addition or subtraction to move all constant terms to the opposite side.
Isolate the Variable: Divide or multiply both sides by the coefficient of the variable to solve for it.
Check Your Solution: Substitute your answer back into the original equation to verify both sides are equal.
Interpret in Context: If the equation came from a word problem, state what the solution means in the context of the original scenario.
Key Formula/Rule:
Properties of Equality:
If a = b, then a + c = b + c (Addition Property)
If a = b, then a − c = b − c (Subtraction Property)
If a = b, then a · c = b · c (Multiplication Property)
If a = b and c ≠ 0, then a / c = b / c (Division Property)
Worked Example:
Problem: A phone plan charges a $25 monthly fee plus $0.10 per text message. Last month, the total bill was $43. How many text messages were sent?
Set up the equation: Let x = number of text messages. Total cost = monthly fee + cost per text, so 25 + 0.10x = 43.
Subtract 25 from both sides: 0.10x = 43 - 25 = 18. This isolates the variable term.
Divide both sides by 0.10: x = 18 / 0.10 = 180.
Check the solution: 25 + 0.10(180) = 25 + 18 = 43. This matches the given total.
Answer: 180 text messages were sent last month.
Why This Matters
Linear equations are the building blocks of algebra. Every time you calculate how long a trip will take at a certain speed, figure out how many hours you need to work to afford something, or convert between measurement systems, you are solving a linear equation. Scientists use them to model relationships, engineers use them to design systems, and economists use them to predict trends. The ability to translate a real-world situation into an equation and solve it is one of the most practical skills in all of mathematics.
A streaming service charges a one-time signup fee of $15 plus $9.50 per month. After several months, Maria has paid a total of $72. How many months has she been subscribed?
Gym A charges $30 per month with no signup fee. Gym B charges $18 per month with a one-time signup fee of $60. After how many months will both gyms cost the same total amount?
A rectangular garden has a length that is 3 meters more than twice its width. If the perimeter of the garden is 54 meters, what is the width of the garden?
A school sold adult tickets for $12 and student tickets for $8 for a concert. They sold a total of 200 tickets and collected $2,040. How many adult tickets were sold?
Jake is saving money for a $340 gaming console. He already has $85 and earns $17 per hour tutoring. He also spends $2 per hour on snacks while tutoring. After working for some hours, he has exactly enough. How many hours did he tutor?
A chemist has 80 mL of a 15% salt solution. How many milliliters of a 40% salt solution must be added so that the resulting mixture is a 25% salt solution?
A small business sells handmade candles. The monthly fixed costs are $450 and each candle costs $3.50 to make. Each candle is sold for $12. How many candles must be sold in a month to break even (total revenue equals total cost)?
A. 53 candles
B. 38 candles
C. 45 candles
D. 60 candles
(E) Mini Quiz - Check Your Understanding!
Mini Quiz
E1. Solve for x: 4x + 10 = 34
A. x = 11
B. x = 8
C. x = 6
D. x = 5
E2. A number is doubled and then 5 is subtracted. The result is 19. What is the number?
A. 7
B. 12
C. 14
D. 10
E3. Solve for x: 3(x - 4) = 2x + 1
A. x = 11
B. x = -11
C. x = 13
D. x = 7
Self-Reflection
How did you feel about this topic?
Confident / Okay / Confused - Circle one
What is one thing you must remember?
Problem-Solving Strategy (4-Step Process)
Challenge Problem: A store is having a sale where every shirt costs the same price and every pair of pants costs the same price. Alex buys 3 shirts and 2 pairs of pants for $86. His friend Bella buys 5 shirts and 2 pairs of pants for $114. Using two equations, find the price of one shirt and one pair of pants. Then determine how much it would cost to buy 4 shirts and 3 pairs of pants.
1. Understand:
2. Plan:
3. Execute:
4. Check:
(F) Answer Key & Detailed Explanations
#
Answer
Type
Solution
1
C
Apply
final_step: The Subtraction Property of Equality justifies this step. Answer: C step_1: Original equation: 3x + 7 = 22. Next step shown: 3x = 15. step_2: The student went from 3x + 7 = 22 to 3x = 15 by subtracting 7 from both sides: 22 - 7 = 15. step_3: Subtracting the same value from both sides of an equation is the Subtraction Property of Equality.
2
A
Apply
final_step: x = 7. Answer: A step_1: Start with 5x - 9 = 26. step_2: Add 9 to both sides: 5x = 26 + 9 = 35. step_3: Divide both sides by 5: x = 35 / 5 = 7. step_4: Check: 5(7) - 9 = 35 - 9 = 26. Correct.
3
D
Apply
final_step: The temperature is 20°C. Answer: D step_1: Substitute F = 68 into F = (9/5)C + 32: 68 = (9/5)C + 32. step_2: Subtract 32 from both sides: 36 = (9/5)C. step_3: Multiply both sides by 5/9: C = 36 × (5/9) = 180/9 = 20. step_4: Check: (9/5)(20) + 32 = 36 + 32 = 68°F. Correct.
4
B
Apply
final_step: Maria has been subscribed for 6 months. Answer: B step_1: Let m = number of months. Total cost equation: 15 + 9.50m = 72. step_2: Subtract 15 from both sides: 9.50m = 57. step_3: Divide both sides by 9.50: m = 57 / 9.50 = 6. step_4: Check: 15 + 9.50(6) = 15 + 57 = 72. Correct.
5
C
Apply
final_step: Both gyms cost the same after 5 months. Answer: C step_1: Gym A total after m months: 30m. Gym B total after m months: 60 + 18m. step_2: Set equal: 30m = 60 + 18m. step_3: Subtract 18m from both sides: 12m = 60. step_4: Divide both sides by 12: m = 5. step_5: Check: Gym A at 5 months = 30(5) = $150. Gym B at 5 months = 60 + 18(5) = 60 + 90 = $150. Equal.
6
A
Apply
final_step: The width of the garden is 8 meters. Answer: A step_1: Let w = width. Length = 2w + 3. step_2: Perimeter formula: 2(length + width) = 54, so 2(2w + 3 + w) = 54. step_3: Simplify inside parentheses: 2(3w + 3) = 54. step_4: Distribute: 6w + 6 = 54. step_5: Subtract 6: 6w = 48. Divide by 6: w = 8. step_6: Check: Width = 8, Length = 2(8) + 3 = 19. Perimeter = 2(8 + 19) = 2(27) = 54. Correct.
7
C
Apply
final_step: 110 adult tickets were sold. Answer: C step_1: Let a = number of adult tickets. Student tickets = 200 - a. step_2: Revenue equation: 12a + 8(200 - a) = 2040. step_3: Distribute: 12a + 1600 - 8a = 2040. step_4: Combine like terms: 4a + 1600 = 2040. step_5: Subtract 1600: 4a = 440. Divide by 4: a = 110. step_6: Check: 110 adult + 90 student = 200 tickets. Revenue: 12(110) + 8(90) = 1320 + 720 = 2040. Correct.
8
B
Apply
final_step: Jake tutored for 17 hours. Answer: B step_1: Jake earns $17/hour but spends $2/hour, so net earning = $15/hour. step_2: Equation: 85 + 15h = 340, where h = hours tutored. step_3: Subtract 85 from both sides: 15h = 255. step_4: Divide both sides by 15: h = 255 / 15 = 17. step_5: Check: 85 + 15(17) = 85 + 255 = 340. Correct.
9
D
Apply
final_step: 53.3 mL of the 40% solution must be added. Answer: D step_1: Let x = mL of 40% solution to add. Salt in 15% solution: 0.15(80) = 12 mL. step_2: Salt in 40% solution: 0.40x. Total mixture: (80 + x) mL at 25%. step_3: Equation: 12 + 0.40x = 0.25(80 + x). step_4: Expand right side: 12 + 0.40x = 20 + 0.25x. step_5: Subtract 0.25x from both sides: 12 + 0.15x = 20. step_6: Subtract 12: 0.15x = 8. Divide by 0.15: x = 8 / 0.15 = 53.33... ≈ 53.3 mL. step_7: Check: Total salt = 12 + 0.40(53.3) = 12 + 21.33 = 33.33. Total volume = 80 + 53.3 = 133.3. Concentration = 33.33/133.3 ≈ 0.25 = 25%. Correct.
10
A
Apply
final_step: The business must sell 53 candles to break even. Answer: A step_1: Let n = number of candles sold. Revenue = 12n. Total cost = 450 + 3.50n. step_2: Break-even equation: 12n = 450 + 3.50n. step_3: Subtract 3.50n from both sides: 8.50n = 450. step_4: Divide both sides by 8.50: n = 450 / 8.50 = 52.94... step_5: Since you cannot sell a fraction of a candle, round up to 53 candles. step_6: Check: Revenue = 12(53) = $636. Cost = 450 + 3.50(53) = 450 + 185.50 = $635.50. At 53 candles, revenue exceeds cost, confirming break-even.
(G) Mini Quiz Answers
Multiple Choice:
E1. C
E2. B
E3. C
Self-Reflection:
I feel good about solving basic two-step equations, but I need more practice with equations that have variables on both sides. I also want to remember to always check my solution by substituting it back into the original equation, because I sometimes make sign errors when moving terms across the equals sign.
Problem-Solving Strategy (4-Step):
1. Understand: A school sold adult tickets at $12 each and student tickets at $8 each. The total number of tickets sold was 200, and the total revenue was $2,040. We need to find how many adult tickets were sold. We have two unknowns but can express one in terms of the other since the total is 200.
2. Plan: Let a = number of adult tickets. Then student tickets = 200 - a. Set up a revenue equation: 12a + 8(200 - a) = 2040. Solve for a using distribution and combining like terms.
4. Check: If 110 adult tickets were sold, then 200 - 110 = 90 student tickets were sold. Revenue = 12(110) + 8(90) = 1320 + 720 = $2,040. This matches the given total revenue, confirming our answer.
(H) Error Analysis Notebook (Growth Mindset Tool)
Problem #
My Score
Error Type
What I Learned / How to Improve
#9
0/15
Concept
"I forgot to check my answer. Next time I will verify my solution."