Which manipulations to this equation maintain its balance?
[tex]3 x+3=12[/tex]
Drag each tile to the correct location on the table.
Balanced
Not Balanced
[tex]3 x+3=12+3[/tex] [tex]\frac{3 x+3}{3}=\frac{12}{3}[/tex] [tex]\frac{1}{3}(3 x+3)=\frac{1}{3}(12)[/tex] [tex]\frac{3 x}{3}+3=\frac{12}{3}[/tex]
[tex]3 x+3-3=12-3[/tex]
Answer (2)
Adding the same value to both sides maintains balance.
Subtracting the same value from both sides maintains balance.
Multiplying both sides by the same value maintains balance.
Dividing both sides by the same non-zero value maintains balance.
Only 3 3 x + 3 = 3 12 does not maintain the balance.
Explanation
Analyzing the Problem We are given the equation 3 x + 3 = 12 and asked to determine which manipulations maintain the balance of the equation. A manipulation maintains the balance if it performs the same operation on both sides of the equation. We will analyze each manipulation individually.
Manipulation 1
3 x + 3 = 12 + 3 : This manipulation adds 3 to both sides of the original equation. Since the same operation is performed on both sides, the balance is maintained.
Manipulation 2
3 3 x + 3 = 3 12 : This manipulation divides both sides of the original equation by 3. Since the same operation is performed on both sides, the balance is maintained.
Manipulation 3
3 1 ( 3 x + 3 ) = 3 1 ( 12 ) : This manipulation multiplies both sides of the original equation by 3 1 . Since the same operation is performed on both sides, the balance is maintained.
Manipulation 4
3 3 x + 3 = 3 12 : This manipulation divides only the 3 x term on the left side by 3, but not the constant term 3. The right side is divided by 3. This is equivalent to dividing the left side by 3 incorrectly. Therefore, this manipulation does not maintain the balance.
Manipulation 5
3 x + 3 − 3 = 12 − 3 : This manipulation subtracts 3 from both sides of the original equation. Since the same operation is performed on both sides, the balance is maintained.
Conclusion Therefore, the manipulations that maintain the balance are: 3 x + 3 = 12 + 3 , 3 3 x + 3 = 3 12 , 3 1 ( 3 x + 3 ) = 3 1 ( 12 ) , and 3 x + 3 − 3 = 12 − 3 . The manipulation that does not maintain the balance is 3 3 x + 3 = 3 12 .
Examples
When solving equations, maintaining balance is like keeping a scale level. If you add or subtract weight from one side, you must do the same to the other to keep it balanced. This principle is used in many real-world scenarios, such as balancing budgets, mixing ingredients in recipes, or calculating forces in physics. Understanding how to manipulate equations while maintaining balance is crucial for solving problems accurately and efficiently.
The manipulations that maintain the balance of the equation 3 x + 3 = 12 are adding or subtracting the same value on both sides, and dividing or multiplying both sides by the same non-zero value. The only manipulation that does not maintain balance is 3 3 x + 3 = 3 12 , since it changes only part of the left side. Therefore, it's important to apply the same operation uniformly across both sides of the equation to ensure it remains balanced.
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