Problem 107 - Elementary Algebra

Question

Simplify. $$ -5(2)-72 $$

Step-by-Step Solution

Answer

The answer is \(-82\).

Step 1: Evaluate the Product

Start by multiplying the numbers inside the parenthesis and the number outside. Calculate \[-5 imes 2 = -10\].

Step 2: Simplify the Expression

Now, take the result from Step 1 and substitute it into the expression. The expression becomes \[-10 - 72\].

Step 3: Combine the Numbers

Add the two numbers to get the final simplified result. Calculate \[-10 - 72 = -82\].

Key Concepts

Simplification

Simplification in algebra involves making an expression shorter and easier to work with. The goal is to express the same mathematical idea in a simpler form without changing its value.
To simplify an expression, you perform all possible arithmetic operations and combine like terms.
In the given exercise, we start by simplifying the expression \(-5(2)-72\). This involves two main steps: computing the multiplication and then performing the subtraction.

• First, solve the multiplication within the parentheses.
• Then, carry out the subtraction with the result from the multiplication and other numbers outside the parentheses.

Following these steps helps in breaking down complex expressions and reaching the simplest form. This approach clarifies calculations and reduces errors in algebraic operations.

Parentheses

In algebra, parentheses are used to group numbers or terms together, and they indicate the order in which operations should be performed.
They are crucial in determining which computations need to be done first.
When you see expressions like \(-5(2)-72\), the numbers inside the parentheses should be handled before any other operations.
This process is governed by the PEMDAS/BODMAS rules: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

• Multiply \(5\) and \(2\) first because they are enclosed in parentheses.
• By addressing the parentheses first, you ensure accurate calculations.

So, always be vigilant about the role of parentheses to maintain the integrity of mathematical expressions.

Negative Numbers

Negative numbers are numbers that are less than zero, and they are represented with a minus sign.Algebra often involves operations with negative numbers, which require careful attention to signs.
In expressions like \(-5(2)-72\), dealing correctly with negative numbers is crucial.

• When multiplying two numbers where at least one of them is negative, the product will be negative.
• Likewise, adding or subtracting negative numbers affects the sign of the result.

To solve \(-10 - 72\), consider it as \(-10 + (-72)\). Both numbers are negative, and combining them maintains the negative outcome, resulting in \(-82\).
Understanding how negative numbers behave in operations helps prevent mistakes and ensures your solutions are correct.

Multiplication

Multiplication is a fundamental operation in algebra where one number is added to itself a certain number of times.
This operation becomes slightly complex when dealing with negative numbers. The sign of the product needs special attention.
In our problem, \(-5\) is multiplied by \(2\), resulting in \(-10\).

• Multiplying a negative number by a positive number always yields a negative product.
• This is because you are effectively subtracting (opposite of adding) the number from zero multiple times.

Once you achieve the multiplication result, you can then use this product in further simplification steps.
By clearly understanding the rules and effects of multiplication, handling algebraic expressions becomes more intuitive and manageable.