Problem 19 - Precalculus Mathematics for Calculus

Question

Find the sum, difference, or product. $$8(2 x+5)-7(x-9)$$

Step-by-Step Solution

Answer

\(9x + 103\)

Step 1: Expand the Expressions

To simplify the expression, start by expanding each bracket. For the first term, multiply 8 by each term inside the parentheses: \[ 8(2x + 5) = 8 \times 2x + 8 \times 5 = 16x + 40. \]For the second term, apply the negative sign and distribute 7: \[ -7(x - 9) = -7 \times x + 7 \times 9 = -7x + 63. \]

Step 2: Combine Like Terms

Now, combine the like terms from the expanded expression. Start by combining the \(x\) terms:\[ 16x - 7x = 9x. \]Next, combine the constant terms:\[ 40 + 63 = 103. \]So the simplified expression is:\[ 9x + 103. \]

Key Concepts

Expanding Brackets

In algebra, expanding brackets is an essential skill when dealing with algebraic expressions. It involves eliminating the parentheses by multiplying each term inside the brackets by the factor outside. Let's consider our example:

• The expression is written as: \(8(2x + 5) - 7(x - 9)\).
• When expanding, you start with \(8(2x + 5)\). Multiply 8 by both \(2x\) and 5: \(8 \times 2x = 16x\) and \(8 \times 5 = 40\).
• Apply the same rule to \(-7(x - 9)\): multiply \(-7\) by both \(x\) and \(-9\), resulting in \(-7x\) and \(+63\).

By expanding the brackets, you convert the expression from a complicated, bracketed form to a simpler, linear one, making it easier to work with. In our case, once brackets are expanded, the expression becomes \(16x + 40 - 7x + 63\). This sets the stage for the next step, combining like terms.

Like Terms

In algebraic expressions, 'like terms' are terms that have the same variables raised to the same power. Identifying and combining these terms is crucial for simplifying expressions. Let's delve into this with our example:

• After expanding the brackets, our expression is \(16x + 40 - 7x + 63\).
• "Like terms" means the terms with the same variable part. Here, \(16x\) and \(-7x\) are like terms as they both contain the variable \(x\) raised to the first power.

To combine like terms, simply add or subtract their coefficients, the numerical parts in front of the variables:

• Combine \(16x\) and \(-7x\): \(16x - 7x = 9x\).
• Constant terms like 40 and 63 can be added together: \(40 + 63 = 103\).

Recognizing and simplifying like terms reduces expressions into a simpler form, in this case: \(9x + 103\).

Simplifying Expressions

Simplifying expressions involves making them easier to read and work with while maintaining their original value. It's a crucial process for solving algebraic problems efficiently. Let's revisit our simplified expression: \(9x + 103\).

• After expanding brackets and combining like terms, the expression becomes much simpler to handle compared to its original form.
• A simplified expression is less cluttered and provides a clearer picture of what you're working with. Instead of complicated steps, you now have \(9x + 103\), which can easily be used in further calculations or problem-solving contexts.

Simplification makes it easier to understand the relationship between different components of an expression and helps identify potential areas for further calculations.

Distributive Property

The distributive property is a fundamental concept in algebra. It allows you to multiply a term across a sum or difference inside parentheses. Mastering this property is key to understanding how algebraic expressions are manipulated.

• The distributive property states that \(a(b + c) = ab + ac\). You distribute the outside term to each inside term.
• In our example, we applied this when expanding \(8(2x + 5)\) to \(16x + 40\) and \(-7(x - 9)\) to \(-7x + 63\).
• It's important to remember to distribute any negative signs as well, which we demonstrated with \(-7(x - 9)\).

By mastering the distributive property, you can simplify and solve more complex expressions efficiently, providing a solid foundation for tackling algebraic challenges.