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Interest calculation is a key concept when it comes to investment word problems. In these problems, interest is typically the money earned from the amount invested over a certain period. There are different types of interests, like simple and compound, but we'll focus on simple interest here since it's what the problem involves.

For Toby's problem, the interest formula used is for simple interest, which is:

• Interest = Principal × Rate × Time

The principal is the initial investment, the rate is the percentage earned per year, and time is the period of investment, usually in years.

In Toby's case, the interest from each account is calculated with this formula. For the 5% investment, his interest is calculated as such:

• Interest from 5% = 0.05 × Principal (x in this case)
• Interest from 7% = 0.07 × Principal (which is 2x for the higher rate)

Adding up these interests gives the total interest Toby earned. Understanding this calculation helps us see how different interest rates affect the total earnings over the same period.

Linear equations are mathematical expressions that describe a straight line when graphed. They are crucial in solving problems where variables need to be determined. In Toby's investment scenario, the solution involves setting up and solving a linear equation to find out the amounts invested.

Here, the linear equation represents the total interest earned from both investments. This is built from the two individual interest expressions for each account:

• For the 5% account: Interest = 0.05x
• For the 7% account: Interest = 0.07(2x)

Adding these expressions, the total interest equation is:\(0.05x + 0.14x = 665\)This equation includes the term for both investments, reflecting Toby's total earnings from the interest. Setting up this equation shows how different rates and amounts affect the total return and helps to unravel the initial amounts invested.

The final step in finding how much Toby invested in each account is solving the equation we set up. Solving equations is about isolating the variable to find its value. The key here is to perform operations that make the equation simpler without changing its equality.

In this problem, we start with the equation:\(0.05x + 0.14x = 665\)We combine like terms to simplify it:

• \(0.19x = 665\)

The objective is to isolate \(x\), the amount invested at 5%. We do this by dividing both sides of the equation by 0.19:

• \(x = \frac{665}{0.19}\)

After calculating, we find \(x \approx 3500\). This implies Toby invested $3500 in the 5% account. For the 7% account, since it is double, he invested \(2 \times 3500 = 7000\). Understanding each step in solving equations ensures the solution makes sense and is correct.