You had 0,000 to invest. You put x dollars in a safe,...

You had $\$10,000$ to invest. You put x dollars in a safe, government-insured certificate of deposit paying 5% per year. You invested the remainder of the money in noninsured corporate bonds paying 12% per year. Your total interest earned at the end of the year is given by the algebraic expression $$0.05 x+0.12(10,000-x)$$ a. Simplify the algebraic expression. b. Use each form of the algebraic expression to determine your total interest earned at the end of the year if you invested $6000 in the safe, government-insured certificate of deposit.

You had $\$10,000$ to invest. You put x dollars in a safe, government-insured certificate of deposit paying 5% per year. You invested the remainder of the money in noninsured corporate bonds paying 12% per year. Your total interest earned at the end of the year is given by the algebraic expression $$0.05 x+0.12(10,000-x)$$
a. Simplify the algebraic expression. b. Use each form of the algebraic expression to determine your total interest earned at the end of the year if you invested $6000 in the safe, government-insured certificate of deposit.

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Key Concepts

Distributive Property

The distributive property in algebra allows you to multiply a single term across terms inside parentheses. This property is essential when simplifying expressions by distributing multiplication over addition or subtraction, making it easier to combine terms and reduce the expression to a simpler form.

Combining Like Terms

Combining like terms is the process of simplifying an algebraic expression by summing or subtracting coefficients of similar terms (terms that contain the same variables raised to the same powers). This technique reduces the complexity of expressions and is a fundamental step in algebraic manipulation.

Substitution Method

The substitution method involves replacing a variable in an algebraic expression with a specific value to evaluate the expression. This method is used to determine the numerical result of an expression once the value of the variable is known, making it especially useful for solving real-world problems.

Linear Functions in Financial Contexts

Linear functions model relationships where the rate of change is constant, which is common in financial scenarios such as calculating interest. By expressing interest earnings as a linear function of the invested amount, one can easily analyze and predict the financial outcome based on different investment strategies.

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