Princeton Review

 Lesson Plan: Solving Linear Equations 

Lesson Objectives

  • Solve linear equations using algebraic manipulation
  • Solve linear equations using graphing
  • Solve linear equations using technology
  • Interpret solutions in real-world contexts

FL BEST Standards

  • MA.8.AR.2.2: Solve two-step linear equations in one variable.
  • MA.8.AR.2.3: Determine slope from table, graph, or description.
  • MA.8.AR.3.2: Write equation in slope-intercept form from table, graph, or description.
  • MA.8.AR.3.5: Solve real-world problems with linear equations in two variables.
  • MA.7.AR.1.1: Add and subtract linear expressions with rational coefficients.
  • MA.7.AR.1.2: Determine if two linear expressions are equivalent.
  • MA.7.AR.2.1: Write and solve one-step equations in one variable.

Prerequisite Skills

  • Understanding of algebraic expressions and equations
  • Familiarity with inverse operations (addition/subtraction, multiplication/division)
  • Combining like terms 

Key Vocabulary

  • Linear equation
  • Solution
  • Inverse operation
  • Like terms

Warm-up Activity (5 minutes)

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.

Teach (20 minutes)

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:

https://www.media4math.com/library/slideshow/slide-show-math-examples-solving-one-step-equations-using-properties-equality

 The examples shown use the following properties of equations:

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:

https://www.media4math.com/library/slideshow/slide-show-math-examples-solving-two-step-equations-using-properties-equality

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

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 hte later lessons.

Review (10 minutes)

Review solving one-step equations by referring to these videos:

Review solving two-step equations by referring to these videos:

Assess (10 minutes)

Administer a 10-question quiz to assess students' understanding of solving linear equations. Include a mix of symbolic equations and real-world problems.

Quiz

  1. Solve for x: x + 5 = 17

     
  2. Solve for y: 2y - 9 = 7

     
  3. Solve for z: 4z + 3 = 19

     
  4. Solve for x: 6x - 2 = 22

     
  5. Solve for y: 5y + 7 = 32

     
  6. John is 5 years older than his sister. If the sum of their ages is 25, how old is John?

     
  7. A baker has 24 cups of flour. If each loaf of bread requires 3 cups of flour, how many loaves can the baker make?

     
  8. 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?

     
  9. 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.

     
  10. 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

  1. x = 12
  2. y = 8
  3. z = 4
  4. x = 4
  5. y = 5
  6. John is 15 years old.
  7. The baker can make 8 loaves of bread.
  8. x - 6 = 42; x = 48
  9. 2x + 75 = 375; girls = 225
  10. The customer bought 2 t-shirts.

 

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