Lesson Plan: Solving Linear Equations
Lesson Summary
This lesson is the third in the middle school sequence on Linear Equations. In this lesson, students learn essential strategies for solving one-variable linear equations by applying fundamental algebraic principles. The lesson focuses on isolating the variable, balancing equations, and verifying solutions through systematic steps. By working through guided examples and interactive activities, students build confidence in solving equations and understanding the logic behind each step. This lesson is designed to reinforce algebraic reasoning and problem-solving skills, aligning with Common Core Standards. It provides a solid foundation for more complex algebraic concepts and prepares students for real-world applications of linear equations.
Lesson Objectives
- Solve linear equations using algebraic methods (e.g., inverse operations, combining like terms)
- Verify solutions to linear equations
- Solve real-world problems involving linear equations
Common Core Standards
- 8.EE.C.7: Solve linear equations in one variable.
- 8.EE.C.7a: Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions.
Prerequisite Skills
- Understanding of algebraic expressions and equations
- Familiarity with inverse operations (addition/subtraction, multiplication/division)
- Combining like terms
Key Vocabulary
- Linear Equation: An algebraic equation that forms a straight line when graphed, typically written in the form y=mx+b or other equivalent formats.
- Variable: A symbol (commonly x or y) that represents an unknown value in an equation.
- Coefficient: A number that multiplies a variable, indicating the variable's factor in the equation.
- Constant: A fixed numerical value in an equation that does not change.
- Solution: A value that, when substituted for the variable, makes the equation true.
- Isolate: The process of rearranging an equation so that the variable appears alone on one side, making it easier to solve.
Multimedia Resources
- Math definitions of terms related to linear equations and functions: https://www.media4math.com/Definitions--LinearFunctions
- Video definitions on the topic of linear equations and functions: https://www.media4math.com/MathVideoCollection--LinearFunctionsDefinitions
Warm Up Activities
Choose from one or more of these activities.
Activity 1: Desmos Activity
Introduce the following Desmos activity to your students:
https://www.desmos.com/calculator/2ws69pq9kc
This graph tracks the wages earned for hourly rates of \$15 to \$25 per hour. Ask students to define the equation for a worker earning $20/hr:
y = 20x
How many hours does this worker need to work to earn 300? This is the linear equation they need to solve:
300 = 20x
or
20x = 300
Ask students how they would solve this equation. Repeat this activity with different hourly wages and different amounts earned. Show that in each case a similar type of linear equation needs to be solved.
Activity 2: Review of Equation Properties
Review the following properties of equations:
- Addition Property of Equality: https://www.media4math.com/library/74671/asset-preview
- Subtraction Property of Equality: https://www.media4math.com/library/74702/asset-preview
- Multiplication Property of Equality: https://www.media4math.com/library/74691/asset-preview
- Division Property of Equality: https://www.media4math.com/library/74677/asset-preview
Activity 3: Review of Combining Like Terms
Remind students of the concept of combining like terms. Provide examples where they need to combine like terms before solving the equation, such as 3x + 2x = 15. Use this Quizlet Flash Card set to review combining like terms.
https://www.media4math.com/library/26721/asset-preview
Teach
Introduction to Solving Linear Equations
One-Step Equations. From the Warm-up Activity, we introduce the notion of a one-step equation. Explain that solving linear equations involves using inverse operations to isolate the variable on one side of the equation.
Here is the solution to the one-step equation shown:
20x = 300
Divide both sides of the equation by 20:
x = 15
Use this slide show to show multiple examples of solving one-step equations, including examples that involve all four basic operations:
Two-Step Equations. Continue the discussion of solving linear equations by introducing two-step linear equations. For example, show equations like these:
2x + 3 = 11
4x - 7 = 19.
These types of equations involve two operations to solve.
Use this slide show to show examples of solving two-step equations:
Verifying Solutions
Emphasize the importance of verifying solutions by substituting the value back into the original equation. Demonstrate this process with the examples used earlier. For example, using the equation from the Warm-up activity:
20x = 300
x = 15
Verify:
20•15 = 300
Review the previous one- and two-step equation examples but from the standpoint of verifying solutions.
Real-World Applications
Introduce real-world scenarios that can be modeled using linear equations, such as age problems, distance-rate-time problems, or mixture problems. Guide students through the process of setting up and solving the equations.
Use this slide show to show a distance-vs-time graph as an application of linear equations and functions:
https://www.media4math.com/library/slideshow/application-linear-equations-distance-vs-time
Emphasize these points:
- The slope of the linear graph is the speed.
- The y-intercept is the initial distance at time t = 0.
This slide show also includes a description of the linear function, which you could return to in one of the later lessons.
Examples
To solve a linear equation, we follow key algebraic principles: balancing both sides of the equation, isolating the variable, and verifying the solution. Below are six detailed examples, including one-step and two-step equations, with real-world applications.
Example 1: Solving by Addition
Problem: x−7=12
- Add 7 to both sides to isolate x: x−7+7=12+7
- Simplify: x=19
- Verify by substituting x=19 into the original equation: 19−7=12
Example 2: Solving by Multiplication
Problem: x5=8
- Multiply both sides by 5 to cancel the fraction: x=8×5
- Simplify: x=40
- Verify by substituting x=40 into the original equation: 405=8
Example 3: Real-World One-Step Equation
Problem: A rental car company charges \$50 per day. If a customer paid \$250, how many days was the car rented?
- Set up the equation: 50d=250
- Divide both sides by 50 to solve for d: d=25050
- Simplify: d=5
- Verify: 50×5=250, confirming the solution.
Example 4: Solving by Addition and Division
Problem: 3x−4=11
- Add 4 to both sides: 3x=15
- Divide by 3: x=153
- Simplify: x=5
- Verify: 3(5)−4=15−4=11, confirming the solution.
Example 5: Solving by Subtraction and Multiplication
Problem: x4+3=10
- Subtract 3 from both sides: x4=7
- Multiply both sides by 4: x=7×4
- Simplify: x=28
- Verify: 284+3=7+3=10
Example 6: Real-World Two-Step Equation
Problem: A gym charges a \$30 membership fee plus \$10 per class. If a member paid \$80, how many classes did they take?
- Set up the equation: 10c+30=80
- Subtract 30 from both sides: 10c=50
- Divide by 10: c=5010
- Simplify: c=5
- Verify: 10(5)+30=50+30=80.
Review
Key Vocabulary
- Linear Equation: An equation that forms a straight line when graphed, usually written as ax+b=c or in slope-intercept form y=mx+b.
- Variable: A symbol (often x or y) representing an unknown value in an equation.
- Coefficient: A number multiplied by a variable, such as the 3 in 3x.
- Constant: A fixed number in an equation, such as the 5 in x+5=10.
- Solution: The value of the variable that makes the equation true.
- Inverse Operations: Operations that undo each other, such as addition/subtraction or multiplication/division, used to isolate the variable.
Solving Equations
Review solving one-step equations by referring to these videos:
- Solving One-Step Addition Equations: https://www.media4math.com/library/1744/asset-preview
- Solving One-Step Subtraction Equations: https://www.media4math.com/library/1745/asset-preview
- One-Step Multiplication Equations: https://www.media4math.com/library/1746/asset-preview
- One-Step Division Equations: https://www.media4math.com/library/1747/asset-preview
Review solving two-step equations by referring to these videos:
- Solving Two-Step Addition and Multiplication Equations: https://www.media4math.com/library/1748/asset-preview
- Solving Two-Step Multiplication and Subtraction Equations: https://www.media4math.com/library/1749/asset-preview
- Solving Two-Step Division and Addition Equations: https://www.media4math.com/library/1750/asset-preview
- Solving Two-Step Division and Subtraction Equations: https://www.media4math.com/library/1751/asset-preview
Additional Examples
Example 1: One-Step Equation (Subtraction)
Problem: x+9=14
- Subtract 9 from both sides: x=14−9
- Simplify: x=5
- Verify: 5+9=14, confirming the solution.
Example 2: One-Step Equation (Division)
Problem: x6=7
- Multiply both sides by 6: x=7×6
- Simplify: x=42
- Verify: 426=7
Example 3: Two-Step Equation (Addition and Division)
Problem: 4x−8=12
- Add 8 to both sides: 4x=20
- Divide by 4: x=204
- Simplify: x=5
- Verify: 4(5)−8=20−8=12.
Example 4: Two-Step Equation (Multiplication and Subtraction)
Problem: x5+2=6
- Subtract 2 from both sides: x5=4
- Multiply both sides by 5: x=4×5
- Simplify: x=20
- Verify: 205+2=4+2=6.
Quiz
- Solve for x: x + 5 = 17
- Solve for y: 2y - 9 = 7
- Solve for z: 4z + 3 = 19
- Solve for x: 6x - 2 = 22
- Solve for y: 5y + 7 = 32
- John is 5 years older than his sister. If the sum of their ages is 25, how old is John?
- A baker has 24 cups of flour. If each loaf of bread requires 3 cups of flour, how many loaves can the baker make?
- A restaurant has a certain number of tables. They clear 6 tables to make space for a dance floor, leaving 42 tables. Write and solve a question to find the total number of tables?
- A school has 375 students. There are 75 more girls than the number of boys. Write and solve the equation to find the number of girls.
- A store sells t-shirts for \$12 each and hats for \$8 each. If a customer spends \$40, and they bought 2 hats, how many t-shirts did they buy?
Answer Key
- x = 12
- y = 8
- z = 4
- x = 4
- y = 5
- John is 15 years old.
- The baker can make 8 loaves of bread.
- x - 6 = 42; x = 48
- 2x + 75 = 375; girls = 225
- The customer bought 2 t-shirts.