Given a long algebraic equation, what are some strategies that you can use to make simplifying and evaluating the equation more efficient and accurate?
How do inverse operations help solve algebraic equations? Give a real-world scenario where this might be modeled. Help, please!!!!
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When simplifying and evaluating a long algebraic equation, there are several strategies that can make the process more efficient and accurate:
1. Combine like terms: Look for terms that have the same variables raised to the same powers and combine them. For example, in the equation 3x + 4x - 2x, the terms with the variable x can be combined to give 5x.
2. Use the distributive property: If you have a term that is being multiplied by a sum or difference, distribute the multiplication to each term within the parentheses. For example, in the equation 2(3x + 4), you can distribute the 2 to get 6x + 8.
3. Simplify fractions: If there are fractions in the equation, simplify them by finding a common denominator and combining the terms. For example, in the equation 2/3x + 3/4, find a common denominator for 3 and 4, which is 12, and then combine the terms accordingly.
4. Use inverse operations: Inverse operations are operations that "undo" each other. In algebra, when solving equations, you can use inverse operations to isolate the variable you are solving for. For example, if you have the equation 2x + 5 = 15, you can subtract 5 from both sides of the equation to get 2x = 10. Then, you divide both sides by 2 to solve for x, giving you x = 5.
Inverse operations are helpful in solving algebraic equations because they allow you to isolate the variable you are trying to solve for. By performing the opposite operation on both sides of the equation, you can "undo" the operations that were applied to the variable and determine its value.
Here's a real-world scenario where inverse operations might be modeled:
Scenario: You go shopping and want to buy a new pair of shoes. The original price of the shoes is $80, but there is a 20% discount. You want to find the final price after the discount.
1. Start by converting the percentage to a decimal. In this case, 20% becomes 0.20.
2. Multiply the original price by the decimal value of the discount. 80 * 0.20 = 16.
3. Subtract the discount from the original price. 80 - 16 = 64.
The final price of the shoes, after the 20% discount, is $64. In this scenario, inverse operations were used to "undo" the discount by subtracting it from the original price, thus finding the final price.