Problem 49 - Elementary Algebra

Question

Find the value of each of the following expressions. $$ -6+1-7 $$

Step-by-Step Solution

Answer

Answer: -12

Step 1: Arrange the given expression

We're given the expression: $$ -6+1-7 $$ Now, we need to solve this expression step by step.

Step 2: Perform the first arithmetic operation

Let's do the addition first: $$ (-6) + (1) = (-6 + 1) = -5 $$ Now, our expression becomes: $$ -5 -7 $$.

Step 3: Perform the second arithmetic operation

Finally, we need to do the subtraction: $$ -5 - 7 = -5 + (-7) = -12 $$ So, after evaluating the given expression, we find out that the value is: $$ -6 + 1 - 7 = -12 $$

Key Concepts

Integer Operations

Integer operations are the building blocks of arithmetic, involving addition, subtraction, multiplication, and division with whole numbers. Integers include zero, positive whole numbers, and negative numbers, providing a full range for calculations.
When working with integers:

• Always remember that positive and negative numbers behave differently in operations.
• Adding a positive number to a negative number is the same as subtracting the opposite positive number. For example, adding 1 to -6 means moving one step towards zero, giving \(-6 + 1 = -5\).
• Subtracting is essentially adding the opposite. Subtraction such as \(-5 -7\) can be viewed as \(-5 + (-7)\).

It’s crucial to grasp these basics of integer operations to handle more complex arithmetic problems easily.

Order of Operations

The order of operations dictates the sequence in which arithmetic operations should be performed to ensure consistent and correct results. This order is crucial for solving expressions accurately, especially those with mixed operations.When dealing with order of operations, follow this sequence:

• Perform calculations inside parentheses first, from innermost to outermost.
• Then handle exponents or powers, although they are not present in simple integer arithmetic.
• Next, execute multiplication and division from left to right.
• Finally, complete addition and subtraction operations from left to right.

In the given expression \(-6 + 1 - 7\), you only need to follow addition and subtraction from left to right. Knowing and applying the order of operations helps in avoiding errors and achieving the correct solution.

Addition and Subtraction

Addition and subtraction are fundamental arithmetic operations that are straightforward yet can become tricky with negative numbers.When adding:

• If the signs are the same, add the numbers and keep the sign.
• If the signs are different, subtract the smaller number from the larger number and keep the sign of the number with the larger absolute value. This is why \(-6 + 1\) becomes \-5\.

For subtraction:

• Consider subtraction as adding a negative. So \(-5 - 7\) is the same as \(-5 + (-7)\).
• Keep an eye on the signs, as they dictate the operation.

By mastering these rules, managing even complex expressions, like \(-6 + 1 - 7 = -12\), becomes more intuitive.