Problem 26 - Elementary Algebra

Question

Find the value of each of the following expressions. $$ 7(-8) $$

Step-by-Step Solution

Answer

Answer: -56

Step 1: Multiply the numbers

Multiply 7 by -8: $$ 7(-8) = -56 $$ The value of the expression $$7(-8)$$ is -56.

Step 2: Write the equation in standard form

Rearrange and simplify the equation.

Step 3: Apply the solution method

Use factoring, quadratic formula, substitution, or other methods.

Step 4: Verify the solution(s)

Check solutions in the original equation.

Step 5: State the final answer

List all valid solutions.

Key Concepts

Integer Multiplication

Integer multiplication involves multiplying two whole numbers, which can be either positive or negative. Understanding how to multiply integers is crucial, as it lays the groundwork for solving more complex problems in algebra and science. The key rules to remember when multiplying integers are:

• **Positive × Positive = Positive**: When both numbers are positive, their product is positive.
• **Negative × Negative = Positive**: Multiplying two negative numbers results in a positive product. The negatives "cancel out."
• **Positive × Negative = Negative**: If one number is positive and the other is negative, the result is negative.
• **Zero × Any Number = Zero**: No matter what the other number is, multiplying by zero results in zero.

When solving \(7(-8)\), you're applying the positive × negative rule. This means the result is negative. Therefore, \(7(-8) = -56\). Remembering these rules helps avoid mistakes in calculations.

Algebraic Expression

An algebraic expression is a mathematical phrase that can involve numbers, variables (like \(x\) or \(y\)), and operators such as addition, subtraction, multiplication, and division. Learning how to evaluate algebraic expressions is important in algebra because it is the foundation for forming and solving equations.When dealing with multiplication in an algebraic expression, use the multiplication rules learned in integer multiplication. Consider the expression \(7(-8)\). Although it is a simple operation, it follows the principle of understanding how changes in sign affect the outcome.
Here's what to keep in mind:

• When you multiply numbers within an expression, apply the correct sign rule based on their signs.
• In more complex expressions, simplify by following the order of operations: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction.
• Understand that each part of the expression can change the final result if miscalculated.

Working with algebraic expressions trains you to think logically and systematically, which will aid you in all areas of mathematics.

Basic Arithmetic Concepts

Arithmetic is the branch of mathematics dealing with numbers and the basic operations—addition, subtraction, multiplication, and division. These are fundamental skills that you need to master early on.For multiplication, especially when involving negative numbers, here are some basics to review:

• **Commutative Property**: This property states that changing the order of the numbers does not change the result. For instance, \(a \times b = b \times a\).
• **Associative Property**: When multiplying more than two numbers, the way they are grouped does not affect the product. Thus \((a \times b) \times c = a \times (b \times c)\).
• **Distributive Property**: Use this to multiply a single term by a group of terms. For example, \(a(b + c) = ab + ac\).

These concepts not only apply to basic arithmetic but also carry through to more advanced areas of math, illustrating their importance in overall mathematical understanding. They will not only help in solving current problems but also in preparing you for more complicated scenarios later on.