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Solving One Step Linear Equations
Balance Scale Basics
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A one step linear equation is an equation with a variable that can be solved using just one inverse operation. The balance scale model shows that both sides of an equation must stay equal, just like two pans holding the same weight. This idea matters because it builds the foundation for solving more complex equations in algebra, physics, chemistry, and many real world problem situations. When students see equations as balanced relationships, the rules for solving them become more logical and less like memorized tricks.
To solve a one step equation, identify what operation is being done to the variable, then undo it with the inverse operation. If x + 5 = 12, subtract 5 from both sides because subtraction undoes addition. If 3x = 18, divide both sides by 3 because division undoes multiplication. The key is that whatever you do to one side of the equation, you must do to the other side to keep the balance unchanged.
Key Facts
- An equation stays true only if both sides remain equal.
- Addition and subtraction are inverse operations: x + a = b means x = b - a.
- Multiplication and division are inverse operations: ax = b means x = b/a, when a is not 0.
- To solve x - 7 = 15, add 7 to both sides: x = 22.
- To solve x/4 = 9, multiply both sides by 4: x = 36.
- Check a solution by substituting the value back into the original equation.
Vocabulary
- Equation
- A mathematical statement that two expressions are equal.
- Variable
- A symbol, usually a letter, that represents an unknown or changeable number.
- Inverse operation
- An operation that undoes another operation, such as subtraction undoing addition.
- Solution
- A value of the variable that makes an equation true.
- Coefficient
- A number multiplied by a variable, such as 5 in 5x.
Common Mistakes to Avoid
- Changing only one side of the equation is wrong because it breaks the equality shown by the balance scale.
- Using the same operation instead of the inverse operation is wrong because solving requires undoing what happened to the variable.
- Forgetting the sign of a number is wrong because x - 6 = 10 and x + 6 = 10 have different solutions.
- Skipping the check step is risky because substituting the answer back into the original equation is the fastest way to catch arithmetic errors.
Practice Questions
- 1 Solve x + 8 = 21 and check your answer by substitution.
- 2 Solve 5x = 45 and state which inverse operation you used.
- 3 A balance scale shows x + 3 on the left pan and 10 on the right pan. Explain why subtracting 3 from both pans keeps the scale balanced and reveals the value of x.