Balancing Act: Solving Equations with Variables on Both Sides

Algebra 1 Grades High School 8:53 Video
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Lesson Description

Learn how to solve algebraic equations where variables appear on both sides. This lesson emphasizes understanding the underlying principles of equation manipulation through visualization, ensuring a solid grasp of the concepts.

Video Resource

Introduction to solving an equation with variables on both sides | Algebra I | Khan Academy

Khan Academy

Duration: 8:53
Watch on YouTube

Key Concepts

  • Isolating the variable
  • Maintaining equality through operations on both sides
  • Visualizing equations for better understanding

Learning Objectives

  • Students will be able to solve linear equations with variables on both sides.
  • Students will be able to explain the reasoning behind each step in the solving process.
  • Students will be able to check their solutions to ensure accuracy.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the basic principles of solving equations: maintaining balance by performing the same operations on both sides. Briefly discuss the goal of isolating the variable.
  • Video Viewing (10 mins)
    Watch the Khan Academy video 'Introduction to solving an equation with variables on both sides'. Encourage students to take notes on the steps Sal takes and his explanations.
  • Guided Practice (15 mins)
    Work through the example problem from the video (2x + 3 = 5x - 2) step-by-step as a class. Emphasize the 'why' behind each step, connecting it to the concept of maintaining equality. Reiterate the visual representation of the equation.
  • Independent Practice (15 mins)
    Provide students with a set of similar equations to solve independently. Encourage them to check their answers. Offer assistance as needed.
  • Wrap-up (5 mins)
    Review the key concepts and address any remaining questions. Briefly introduce the idea of more complex equations and how the same principles apply.

Interactive Exercises

  • Equation Balance
    Use a virtual balance scale to represent equations. Students can add or subtract terms from both sides of the equation, visually seeing how the balance is maintained. This reinforces the concept of equality.
  • Step-by-Step Solver
    Provide students with a partially completed equation-solving problem. They must fill in the missing steps and justify their choices, demonstrating their understanding of the process.

Discussion Questions

  • Why is it important to perform the same operation on both sides of an equation?
  • What does it mean to 'isolate the variable'?
  • Can you think of a real-world scenario where solving an equation with variables on both sides might be useful?

Skills Developed

  • Algebraic manipulation
  • Problem-solving
  • Logical reasoning

Multiple Choice Questions

Question 1:

What is the first step in solving the equation 3x + 5 = x - 1?

Correct Answer: Subtract x from both sides

Question 2:

When solving an equation, what does it mean to 'isolate the variable'?

Correct Answer: To get the variable by itself on one side of the equation

Question 3:

Solve for x: 4x - 2 = 2x + 6

Correct Answer: x = 4

Question 4:

Which operation maintains the equality of an equation?

Correct Answer: Performing the same operation on both sides

Question 5:

Solve for x: x + 7 = 2x - 3

Correct Answer: x = 2

Question 6:

Which of the following is a valid operation to simplify the equation 5x + 2 = 3x - 4?

Correct Answer: Subtracting 5x from both sides.

Question 7:

If you have the equation 6x - 3 = 2x + 5, what is the next best step?

Correct Answer: Subtract 2x from both sides.

Question 8:

What value of x satisfies the equation 7x + 1 = 4x + 10?

Correct Answer: x = 3

Question 9:

What operation should you perform to isolate 'x' in the equation x/2 + 3 = 7?

Correct Answer: Subtract 3 first.

Question 10:

What is the solution to the equation 9x - 5 = 4x + 15?

Correct Answer: x = 4

Fill in the Blank Questions

Question 1:

The goal of solving an equation is to __________ the variable.

Correct Answer: isolate

Question 2:

To maintain balance in an equation, you must do the same __________ to both sides.

Correct Answer: operation

Question 3:

If 5x + 2 = 3x + 8, then 2x = __________.

Correct Answer: 6

Question 4:

The solution to the equation 4x - 1 = x + 5 is x = __________.

Correct Answer: 2

Question 5:

Before dividing to isolate a variable, you often need to __________ like terms.

Correct Answer: combine

Question 6:

In the equation 8x - 4 = 3x + 6, adding 4 to both sides results in 8x = 3x + __________.

Correct Answer: 10

Question 7:

When solving an equation, any value for x that makes the equation true is called a __________.

Correct Answer: solution

Question 8:

Subtracting 3x from both sides of the equation 5x = 3x + 10 results in 2x = __________.

Correct Answer: 10

Question 9:

To solve for x in the equation x/3 - 1 = 4, you would first add 1 to both sides, then __________ by 3.

Correct Answer: multiply

Question 10:

In the equation 2x + 5 = x + 9, the value of x that makes the equation balanced is __________.

Correct Answer: 4

Educational Standards

A1.A.1:Represent and solve mathematical and real-world problems using linear equations, absolute value equations, and systems of equations; interpret solutions in the original context. A1.A.3:Create and evaluate equivalent algebraic expressions and equations using algebraic properties. A1.A.3.1:Solve equations involving several variables for one variable in terms of the others.

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