Questions and Answers
What characterizes a unique solution in a one-variable equation?
Which equation has no solution?
What is the first step to solve the equation 5x + 4 = 24?
If an equation simplifies to 0 = 0, what type of solution does it have?
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When combining like terms in the equation 3x + 2x + 5 = 20, what is the result?
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Which operation is used to isolate the variable in the equation 4x = 36?
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In the equation 2(x - 3) = 4, what is the first step after distributing?
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What common mistake might one make when solving 2x + 4 = 10?
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Study Notes
Solving One-Variable Equations
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Definition: A one-variable equation is an equation that contains only one variable, typically represented as ( x ). The general form is ( ax + b = c ).
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Types of Solutions:
- Unique Solution: One value for ( x ) satisfies the equation.
- No Solution: The equation is contradictory (e.g., ( x + 2 = x - 3 )).
- Infinite Solutions: All values for ( x ) satisfy the equation (e.g., ( 2x - 2 = 2(x - 1) )).
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Basic Steps to Solve:
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Isolate the Variable:
- Use inverse operations (addition, subtraction, multiplication, division) to get the variable alone on one side.
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Simplify:
- Combine like terms and simplify each side of the equation as needed.
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Check Solution:
- Substitute the obtained value back into the original equation to verify correctness.
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Isolate the Variable:
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Example Problems:
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Example 1: Solve ( 3x + 5 = 20 )
- Subtract 5 from both sides: ( 3x = 15 )
- Divide by 3: ( x = 5 )
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Example 2: Solve ( 2(x - 3) = 4 )
- Distribute: ( 2x - 6 = 4 )
- Add 6: ( 2x = 10 )
- Divide by 2: ( x = 5 )
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Common Mistakes:
- Forgetting to perform the same operation on both sides.
- Misapplying the distributive property.
- Miscalculating when combining like terms.
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Special Considerations:
- If the variable cancels out entirely (e.g., ( 0 = 0 )), the equation has infinite solutions.
- If the result is a false statement (e.g., ( 0 = 5 )), there is no solution.
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Applications:
- Used in various fields such as finance (budgeting, profit calculations), physics (motion problems), and everyday problem-solving scenarios.
Definition and Types of Solutions
- A one-variable equation contains a single variable, typically denoted as ( x ).
- General form of a one-variable equation: ( ax + b = c ).
- Solutions can be categorized as:
- Unique Solution: Only one value for ( x ) satisfies the equation.
- No Solution: The equation has conflicting outcomes (e.g., ( x + 2 = x - 3 )).
- Infinite Solutions: Any value for ( x ) satisfies the equation (e.g., ( 2x - 2 = 2(x - 1) )).
Steps to Solve One-Variable Equations
- Isolate the Variable: Use inverse operations to get ( x ) alone on one side.
- Simplify: Combine like terms and simplify both sides of the equation.
- Check Solution: Substitute the found value back into the original equation to confirm it holds true.
Example Problems
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Problem 1: ( 3x + 5 = 20 )
- Subtract 5 from both sides to get ( 3x = 15 ).
- Divide by 3 to find ( x = 5 ).
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Problem 2: ( 2(x - 3) = 4 )
- Distribute to get ( 2x - 6 = 4 ).
- Add 6 to both sides resulting in ( 2x = 10 ).
- Divide by 2 to determine ( x = 5 ).
Common Mistakes
- Neglecting to perform the same operation on both sides of the equation.
- Misapplying the distributive property during simplification steps.
- Errors in combining like terms leading to incorrect solutions.
Special Considerations
- If the variable cancels completely (e.g., resulting in ( 0 = 0 )), the equation has infinite solutions.
- A false statement (e.g., ( 0 = 5 )) indicates no solutions exist.
Applications
- Relevant in fields like finance for budgeting and profit calculations.
- Used in physics for solving motion problems and other quantitative analyses.
- Applicable in everyday problem-solving scenarios, enhancing analytical skills and logical reasoning.
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Description
Test your knowledge on solving one-variable equations! This quiz covers unique, no, and infinite solutions, along with basic steps to isolate the variable, simplify, and check your solution. Challenge yourself with example problems and see how well you understand the concept.
