Problem 67 - Beginning and Intermediate Algebra

Question

Insert one pair of parentheses in each expression so that the given value results when the operations are performed. $$ \begin{array}{r} 10-7-3 \\ =6 \end{array} $$

Step-by-Step Solution

Answer

Place parentheses: 10 - (7 - 3).

Step 1: Analyze the Expression

Look at the given expression: 10 - 7 - 3 = 6First, identify the operations present. Here, we have two subtraction operations.

Step 2: Determine Correct Parentheses Placement

To achieve the result of 6, place the parentheses to ensure the correct order of operations. Note that parentheses alter the standard order of operations. Place the parentheses around '10 - 7' to evaluate that first:(10 - 7) - 3

Step 3: Evaluate the Expression

Calculate inside the parentheses first:10 - 7 = 3Then, perform the remaining subtraction:3 - 3 = 0This does not equal 6, so try placing the parentheses around '7 - 3' instead:10 - (7 - 3)

Step 4: Verify the Solution

Evaluate the new expression with the correct parentheses placement:First calculate inside the parentheses:7 - 3 = 4Then perform the remaining subtraction:10 - 4 = 6This matches the given value.

Key Concepts

Parentheses Placement

When solving mathematical expressions, understanding where to place parentheses is crucial. Parentheses can change the order in which you perform operations, leading to different results. In our example, the expression is 10 - 7 - 3. Without parentheses, operations are performed from left to right. For instance, calculate:
10 - 7 = 3
Then, 3 - 3 = 0.
This result doesn't match the target value of 6.
Therefore, you need to adjust parentheses placement to change the order.
Instead, try placing them around the second part:
10 - (7 - 3).
This ensures you first calculate inside the parentheses (7 - 3), which gives 4.
Then, 10 - 4 equals 6, matching the given value.
The proper use of parentheses is a powerful tool for controlling the sequence of operations and obtaining the correct result.

Subtraction

Subtraction is one of the basic arithmetic operations and involves taking one number away from another. In our problem, the original expression was 10 - 7 - 3.
Without parentheses, the subtraction operations are performed from left to right:
10 - 7 = 3
Then, 3 - 3 = 0.
However, inserting parentheses can change the order in which these subtractions are executed.
By placing parentheses around '7 - 3', you change the operations to:
First calculate inside the parentheses: 7 - 3, which equals 4.
Then subtract this result from 10:
10 - 4 = 6.
This way, subtraction can yield different results based on the order of operations, which is controlled by parentheses.

Evaluation of Expressions

Evaluating expressions correctly is key in mathematics. It's about following a set of rules, sometimes known as the 'order of operations', to compute a final value.
For the given problem, the expression is 10 - 7 - 3. Evaluating this directly would yield:
10 - 7 = 3
Then, 3 - 3 = 0.
This does not match our required value of 6.
By inserting the correct parentheses, the order of operations changes.
Let's analyze: Place parentheses around '7 - 3' and the new expression becomes:
10 - (7 - 3).
First calculate inside the parentheses: 7 - 3 = 4.
Then, perform the remaining subtraction:
10 - 4 = 6. This result matches the target value.
The key takeaway here is that evaluating expressions correctly often depends on manipulating the order of operations using parentheses, enabling you to reach the desired outcome accurately.