Multi-step Equations

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What is the first step in solving multi-step equations?

Isolate the variable

Move variable terms

Simplify each side (correct)

Move constant terms

In the equation 3(x - 4) + 5 = 10, which operation should be performed first?

Distribute 3 (correct)

Add 5 to both sides

Subtract 4 from x

Subtract 10 from both sides

When solving for x in the equation x/4 + 2 = 6, what is the first step?

Add 2 to both sides

Subtract 2 from both sides (correct)

Divide both sides by 4

Multiply both sides by 4

Which of the following indicates a scenario where there is no solution?

2x + 3 = 2x + 5 Signup and view all the answers

What does it mean when an equation has infinite solutions?

The variables are dependent Signup and view all the answers

Which technique is used to eliminate parentheses in an equation?

Applying the distributive property Signup and view all the answers

Which of the following is the correct solution for x in the equation 5(x + 1) = 30?

x = 5 Signup and view all the answers

What should you do after finding the value of the variable?

Substitute back into the original equation Signup and view all the answers

Why is it important to keep the equation balanced during solving?

To ensure accuracy and correctness Signup and view all the answers

Explain how to isolate the variable in the equation 4x + 6 = 22.

First, subtract 6 from both sides to get 4x = 16, then divide by 4 to find x = 4. Signup and view all the answers

What is the result of solving the equation 6x - 12 = 30?

The solution is x = 7 after adding 12 to both sides and then dividing by 6. Signup and view all the answers

Describe a common mistake when solving two-step equations.

A common mistake is forgetting to perform the same operation on both sides, leading to an incorrect solution. Signup and view all the answers

If you have the equation 2x + 5 = 15, how would you solve for x?

Subtract 5 from both sides to get 2x = 10, then divide by 2 to find x = 5. Signup and view all the answers

What does it mean to check your solution in an equation?

It means substituting the found value back into the original equation to verify both sides are equal. Signup and view all the answers

How do you solve the equation 7x + 3 = 24?

Subtract 3 from both sides to get 7x = 21, then divide by 7 to find x = 3. Signup and view all the answers

What should you do if, while solving an equation, you encounter a negative coefficient?

You should still isolate the variable by performing opposite operations, treating the negative number like any other constant. Signup and view all the answers

In the equation 8 - 2x = 0, how would you solve for x?

First, subtract 8 from both sides to get -2x = -8, then divide by -2 to find x = 4. Signup and view all the answers

Study Notes

Multi-step Equations

Definition: Equations that require more than one operation to isolate the variable.
General Steps to Solve:
1. Simplify Each Side:
  - Combine like terms.
  - Distribute if necessary.
2. Move Variable Terms:
  - Use addition or subtraction to get all variable terms on one side.
3. Move Constant Terms:
  - Use addition or subtraction to get all constant terms on the other side.
4. Isolate the Variable:
  - Use multiplication or division to solve for the variable.
Common Techniques:
- Combining Like Terms: Group terms with the same variable or constants.
- Distributive Property: Apply a(b + c) = ab + ac to eliminate parentheses.
- Inverse Operations: Use opposites to move terms (e.g., + for -).
Example Problem:
- Solve for x: 2(x + 3) = 16
  1. Distribute: 2x + 6 = 16
  2. Subtract 6: 2x = 10
  3. Divide by 2: x = 5
Checking Solutions: Substitute the found variable back into the original equation to verify correctness.
Special Cases:
- No Solution: Occurs when variables cancel out and results in a false statement (e.g., 0 = 5).
- Infinite Solutions: Happens when variables cancel out leading to a true statement (e.g., 0 = 0).
Tips:
- Write each step clearly to avoid mistakes.
- Keep the equation balanced by performing the same operation on both sides.
- Be mindful of negative signs; they can easily lead to errors.

Multi-step Equations

Equations requiring multiple operations to isolate a variable.

General Steps to Solve

Simplify Each Side: Combine like terms and distribute as needed.
Move Variable Terms: Use addition or subtraction to get all variable terms on one side of the equation.
Move Constant Terms: Use addition or subtraction to move all constant terms to the opposite side.
Isolate the Variable: Employ multiplication or division to solve for the variable.

Common Techniques

Combining Like Terms: Group together terms that share the same variable or constant values.
Distributive Property: Apply the rule a(b + c) = ab + ac to simplify expressions by removing parentheses.
Inverse Operations: Utilize opposite operations for term manipulation, such as adding when subtracting.

Example Problem

For the equation 2(x + 3) = 16:
- Distributing yields 2x + 6 = 16.
- Subtracting 6 gives 2x = 10.
- Dividing by 2 results in x = 5.

Checking Solutions

Always substitute the solved variable back into the original equation to confirm it satisfies the equation.

Special Cases

No Solution: Occurs when variable cancelations lead to a false statement, such as 0 = 5.
Infinite Solutions: Happens when variable cancelations yield a true statement, like 0 = 0.

Tips for Solving

Document each step clearly to prevent errors.
Maintain balance in the equation by executing the same operation on both sides.
Be vigilant with negative signs, as they can easily cause mistakes.

Two-step Equations Overview

Definition: Equations requiring two steps to isolate a variable, crucial in algebra.
General Form: Typically written as ax + b = c
- a: Coefficient of the variable x
- b: Constant added/subtracted from the variable
- c: Constant value resulting from the equation

Steps to Solve Two-step Equations

Isolate the Variable Term:
- Move the constant term (b) by adding or subtracting it from both sides of the equation.
- Example: For (2x + 3 = 11), subtract 3 to get (2x = 8).
Solve for the Variable:
- Divide or multiply to isolate x using the coefficient (a).
- Continuing from the previous example, (x = \frac{8}{2}) results in (x = 4).

Example Problems

Solving (3x - 4 = 5):
- Add 4 to both sides: (3x = 9).
- Divide by 3 to isolate x: (x = 3).
Solving (5x + 2 = 17):
- Subtract 2 from both sides: (5x = 15).
- Divide by 5 to find x: (x = 3).

Important Notes

Maintain equality by performing the same operation on both sides of the equation.
Always check solutions by substituting back into the original equation to verify accuracy.
Adhere to the order of operations (PEMDAS/BODMAS) when manipulating equations.

Common Mistakes

Forgetting to perform equal operations on both sides of the equation, leading to incorrect solutions.
Miscalculations during addition, subtraction, multiplication, or division, which can derail problem-solving.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

This quiz covers the concept of multi-step equations, including definitions, general steps to solve, and common techniques used. Learn how to isolate variables and apply the distributive property while solving example problems. Test your understanding of solving linear equations effectively.