Problem 58 - Elementary Algebra

Question

Find the value of each of the following expressions. $$ -9(0-2)+4(8-9)+0(-3) $$

Step-by-Step Solution

Answer

Question: Evaluate the expression: $$(0-2)(-9) + (8-9)4 + (0)(-3)$$ Answer: 14

Step 1: Simplify the expressions inside the parentheses

Start by simplifying the expressions inside the parentheses. Calculate the difference between the numbers inside each parenthesis: $$ 0-2 = -2 \\ 8-9 = -1 $$ Now, the expression looks like this: $$ -9(-2) + 4(-1) + 0(-3) $$

Step 2: Perform the multiplication operations

Next, we will multiply each term by its respective multiplier: $$ -9\times-2 = 18 \\ 4\times-1 = -4 \\ 0\times-3 = 0 $$ Now, the expression looks like this: $$ 18 - 4 + 0 $$

Step 3: Perform the addition and subtraction operations

Lastly, add and subtract the numbers in the expression, from left to right: $$ 18 - 4 = 14 \\ 14 + 0 = 14 $$ The value of the given expression is 14.

Key Concepts

Simplifying Expressions

Simplifying expressions is the first step when tackling any mathematical problem involving parentheses. This step often involves working out what is inside the parentheses first. By doing this, we convert a complex expression into a simpler form that is easier to manage. In this context, we look at expressions like \(0 - 2\) and \(8 - 9\) and solve them, turning them into \(-2\) and \(-1\) respectively.
This simplification is crucial because it lays the groundwork for all further calculations. It ensures that we deal with the straightforward and correct values when conducting operations such as multiplication or addition. It also helps prevent mistakes that can occur if calculations are done out of order. Always remember: simplify inside the parentheses before doing anything else.
There's a clear mental checklist one should use to simplify expressions:

• Identify and solve anything inside parentheses or brackets.
• Reduce expressions to their simplest form by performing any addition or subtraction.
• Replace complex expressions with their simplified versions.

Multiplication

Once expressions have been simplified, the next step is to move on to multiplication. This action involves taking numbers and multiplying them correctly to reach their products. In our exercise, after simplification, the terms look like this: \(-9(-2)\), \(4(-1)\), and \(0(-3)\).
Each of these terms needs to be multiplied to determine its value:

• The first term, \(-9\times-2 = 18\), shows how multiplying two negative numbers results in a positive.
• The second term, \(4\times-1 = -4\), demonstrates how a positive number times a negative results in a negative number.
• The last term \(0\times-3 = 0\), emphasizes the unique property of zero; any number multiplied by zero results in zero.

Using correct multiplication rules will ensure that the expression is transformed correctly and make subsequent calculations easier and more accurate.
Always check your multiplication steps twice, especially when dealing with negative numbers!

Addition and Subtraction

After executing the multiplication steps, our focused attention should turn to addition and subtraction. This is about adding and subtracting values to reach a final simplified result. With our multiplication results, the expression is further simplified to \(18 - 4 + 0\).
This last stage is about performing these operations from left to right, ensuring the sequence of operations remains intact:

• First, calculate \(18 - 4 = 14\).
• Next, incorporate the zero by solving \(14 + 0 = 14\).

This process involves clear sequential calculations. It's beneficial to check each operation to ensure there are no errors in the addition or subtraction, leading to the final solution.
Remember, when operating with basic arithmetic elements, order is paramount as defined by the guidelines called PEMDAS/BODMAS, which refers to parentheses, exponents, multiplication, and division, followed by addition and subtraction.