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Graphs of Linear Systems

Graph lines to identify intersection points

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We all fuss about taxes. Large corporations, as well as individuals, are always looking for ways to save on taxes. How do we get to a lower tax bracket? How do we maximize tax deductions and increase tax credits? Whether we use software, seek an accountant’s help, or do it ourselves, it all comes down to a set of linear relations and equations.

Why It Matters

The goal of tax planning is to minimize your tax burden. If you look at the math, it's a combination of percentages, subtractions, and additions, which means you can calculate anything you want using a set of linear equations.

Besides your income and household size, there are many other factors that need to be taken into consideration, including different sorts of credits and deductions. Though it's true that the tax laws apply to everyone equally, people are free to arrange their financial affairs in order to take advantage of the system. For example, you can reduce your total income as well as your income tax if you contribute money to a retirement account. This might even help you stay in a lower tax bracket.

You can find more on how income taxes are calculated here: http://www.khanacademy.org/science/core-finance/taxes-topic/taxes/v/basics-of-us-income-tax-rate-schedule

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Here’s a seemingly simple tax problem. A corporation has a taxable income of $10 million, on which it must pay federal taxes at a 30% rate. You need to calculate the taxes. Easy, right? That’s $3 million!

But what if we added some state taxes? Suppose that rate is 10%. But the federal taxes are based on the income after the state taxes are deducted, and the state taxes are based on the income after the federal taxes are deducted. What are the two taxes now? How can you use a system of linear equations to find the answer?

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