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Determining the Value of A: Solving for Multiplicative Inverse Equations
Understanding Multiplicative Inverse Equations
Mathematics is a vast domain that involves the manipulation of numbers and symbols to derive meaningful conclusions. One of the fundamental concepts in mathematics is the multiplicative inverse. In this article, we will explore the concept of multiplicative inverse, specifically, how to find the value of A in the equation 5a-7^{-1} frac{1}{7}.
Simplifying the Multiplicative Inverse Equation
The given equation is 5a-7^{-1} frac{1}{7}. To solve for A, let's first understand the meaning of 7^{-1}. An inverse of a number in the context of multiplication is a value that, when multiplied by the original number, yields 1. So, 7^{-1} is the reciprocal of 7, which is frac{1}{7}.
5a - 7 ^{-1} frac{1}{7}
Substituting the reciprocal of 7, we get:
5a - frac{1}{7} frac{1}{7}
Step-by-Step Solution
The equation can be simplified as follows:
Start with the given equation: 5a - frac{1}{7} frac{1}{7} Add frac{1}{7} to both sides to isolate the term with A: 5a frac{1}{7} frac{1}{7} Simplify the right-hand side: frac{1}{7} frac{1}{7} frac{2}{7} So, the equation becomes: 5a frac{2}{7} Multiply both sides by 5 to solve for A: a frac{2}{7} * frac{1}{5} frac{2}{35}Alternative Approach
Another approach to solve the equation is to simplify the steps directly:
Start with the given equation: 5a - 7^{-1} frac{1}{7} Recognize that 7^{-1} frac{1}{7}, so the equation becomes: 5a - frac{1}{7} frac{1}{7} Isolate the term with A: 5a frac{1}{7} frac{1}{7} frac{2}{7} Solve for A: a frac{2}{7} * frac{1}{5} frac{2}{35}Verification
To verify the solution, substitute a frac{2}{35} into the original equation:
Original equation: 5a - 7^{-1} frac{1}{7} Substitute a frac{2}{35}: 5 * frac{2}{35} - 7^{-1} frac{1}{7} Calculate: frac{10}{35} - 7^{-1} frac{1}{7} Simplify: frac{2}{7} - 7^{-1} frac{1}{7} Since 7^{-1} frac{1}{7}, frac{2}{7} - frac{1}{7} frac{1}{7} This confirms that the solution is correct: a frac{2}{35}Conclusion
In conclusion, we have solved the equation 5a - 7^{-1} frac{1}{7} and determined that the value of A is frac{2}{35} or 2.8 (rounded to one decimal place).