91Ó°ÊÓ

A tax formula is an equation used to calculate the amount of tax owed based on certain financial figures. In this particular exercise, we are given a tax formula that involves income and a fixed amount, and we need to solve the equation for the income variable, denoted as \(I\). The formula provided is \(T=\frac{3}{10}(I-12,000)\). This means the tax \(T\) is a fraction of the income after subtracting a specific amount, here 12,000.
Tax formulas are commonly used in financial calculations, helping to determine how much tax one needs to pay. By understanding how to manipulate and solve these equations, you can gain insights into how changes in income affect taxes.

Isolating a variable means rearranging an equation so that the variable you're solving for stands alone on one side of the equation. This process usually involves a series of steps where you reverse the mathematical operations affecting the variable.
In our example, to isolate \(I\) in the equation \(T=\frac{3}{10}(I-12,000)\), we need to perform operations that will leave \(I\) by itself on one side. This involves eliminating the fraction and constants. Here's a step-by-step breakdown:

• Remove the fraction: Multiply both sides by the reciprocal of \(\frac{3}{10}\), which is \(\frac{10}{3}\).
• Perform the multiplication: This simplifies the equation to \[\frac{10}{3} \cdot T = I - 12,000\]
• Solve for \(I\): Finally, add 12,000 to both sides to completely isolate \(I\): \[\frac{10}{3} \cdot T + 12,000 = I\]

By systematically performing these operations, we can successfully isolate \(I\) and solve the equation.

The reciprocal of a number is what you multiply that number by to get a product of 1. For example, the reciprocal of 5 is \(\frac{1}{5}\), and the reciprocal of \(\frac{3}{10}\) is \(\frac{10}{3}\).
Reciprocals are fundamental in solving equations involving fractions. They allow you to clear fractions efficiently by converting them into simpler forms. In our exercise, to remove the fraction \(\frac{3}{10}\) from the left side of the equation, we multiply both sides by its reciprocal, \(\frac{10}{3}\):

• Given: \[T=\frac{3}{10}(I-12,000)\]
• Multiply by \(\frac{10}{3}\): \[\frac{10}{3} \cdot T = \frac{10}{3} \cdot \frac{3}{10} (I-12,000)\]

This simplifies the fraction on the right side to 1, leaving us with a cleaner equation to work with: \[\frac{10}{3} \cdot T = I - 12,000\]
Understanding reciprocals makes it easier to handle fractional coefficients when solving equations.