Problem 63 - Algebra for College Students

Question

Simplify each numerical expression. $$-5+(-2)(7)-(-3)(8)$$

Step-by-Step Solution

Answer

The simplified expression is 5.

Step 1: Understand the Expression

The expression to simplify is \(-5 + (-2)(7) - (-3)(8)\). This contains addition, subtraction, and multiplication operations, as well as negative numbers, so we need to follow the order of operations (PEMDAS/BODMAS: Parentheses/Brackets, Orders/Exponents, Multiplication and Division, Addition and Subtraction).

Step 2: Perform Multiplications

Calculate the products of the numbers inside the parentheses first:1. \((-2)(7) = -14.\)2. \(-(-3)(8) = +24.\) Note that multiplying by a negative number changes the sign.

Step 3: Substitute Back into the Expression

Replace the original products with the results from Step 2:\(-5 - 14 + 24\).

Step 4: Simplify Addition and Subtraction

To solve \(-5 - 14 + 24\), perform the operations from left to right:1. Add \(-5 - 14 = -19.\)2. Add the result to 24: \(-19 + 24 = 5.\)

Step 5: Write the Final Answer

The simplified expression is \(5.\)

Key Concepts

Order of Operations

To simplify any mathematical expression correctly, it is crucial to follow the Order of Operations. This set of rules dictates the sequence in which different operations should be performed. For any numerical expression, the sequence typically follows the acronym PEMDAS:

• Parentheses - Deal with expressions inside parentheses or brackets first.
• Exponents - Solve exponentials or powers, if any.
• Multiplication and Division - Perform these operations from left to right.
• Addition and Subtraction - Finally, tackle these operations from left to right.

Using our example -5 + (-2)(7) - (-3)(8), the first operation we perform is multiplication since there are no parentheses or exponents to deal with. Observing the correct order ensures consistency and accuracy, avoiding common pitfalls like performing operations in the wrong sequence.

Negative Numbers

Negative numbers can seem tricky, but understanding how they interact during mathematical operations is essential. Consider negative numbers in terms of value direction; they indicate a move below zero on a number line.

When working with negative numbers, note the following:

• Adding a negative number is equivalent to subtraction. For instance, -5 + (-3) can be seen as -5 - 3.
• Subtracting a negative number results in addition: -7 - (-2) is the same as -7 + 2.
• Multiplying two negative numbers yields a positive product. Conversely, multiplying a positive number with a negative one results in a negative product.

In our example, -5 + (-2)(7) - (-3)(8), notice how multiplication of negative numbers affects the sign of their product, turning -(-3)(8) into a positive 24.

Multiplication in Algebra

In algebra, multiplication often involves variables and constants, and it's vital to maintain clarity regarding the rules that govern these operations. While dealing with multiplication in expressions, consider these points:

- Multiplying numbers involves regular arithmetic computation; however, signs must be observed carefully.
- If you multiply two positive numbers or two negative numbers, the result is positive.
- Multiplying a positive number by a negative number, results in a negative product.

In the expression -5 + (-2)(7) - (-3)(8), performing the multiplication first, while considering the signs, simplifies the remaining calculations. This involves computing: * (-2)(7) resulting in -14, and * -(-3)(8) resulting in +24.

Correctly handling multiplication of negative numbers at this step is crucial for accurate simplification of algebraic expressions.