Problem 7 - Algebra 1

Question

Evaluate the expression for the given value of the variable. $$\frac{16}{x}-2 \text { when } x=4$$

Step-by-Step Solution

Answer

The evaluated expression is 2.

Step 1: Substitution

Substitute the given value, x = 4, into the expression. So the expression becomes \( \frac{16}{4} - 2 \)

Step 2: Perform division

Perform the division operation, \( \frac{16}{4} \). This evaluates to 4. So the expression now becomes 4 - 2.

Step 3: Perform subtraction

Now subtract 2 from 4 to evaluate the expression. This gives the solution 2.

Key Concepts

Variable Substitution

Variable substitution is a foundational technique in algebra that involves replacing a variable with its corresponding numerical value. The purpose of this approach is to simplify an algebraic expression to a point where it can be easily solved. In the given example, we are asked to evaluate the expression \( \frac{16}{x} - 2 \) for the given value of the variable, which is x=4.

During variable substitution, it's crucial to replace every occurrence of the variable in the expression with the given number. This direct replacement process helps in transforming an algebraic expression into an arithmetic one, making it solvable using basic operations like addition, subtraction, multiplication, and division. Here's a simple rule to follow for variable substitution: make sure to write down the new numerical expression clearly after substituting to avoid any mistakes in further steps of calculation.

Division Operation

The division operation is an arithmetic process of determining how many times one number is contained within another. When dealing with an algebraic expression that includes division, it is often one of the first operations that needs to be performed after variable substitution. In our case, after substituting the value of x with 4, the expression becomes \( \frac{16}{4} - 2 \). The division here indicates that 16 is to be divided by 4.

To properly execute the division operation, divide the numerator (the top number) by the denominator (the bottom number). It is also helpful to know that division by a non-zero number can be performed, while division by zero is undefined and must be avoided. The result of the division operation will then be used in subsequent steps, such as further arithmetic operations, to finally evaluate the expression.

Arithmetic Subtraction

Arithmetic subtraction is the process of taking away one number from another to find the difference. It is one of the four basic operations in mathematics and is used to solve algebraic expressions that involve a minus sign. After performing the division operation in the expression \( \frac{16}{4} - 2 \), we obtain a simplified form of the expression: 4 - 2.

Subtraction is performed by taking the second number (the subtrahend) from the first number (the minuend). In our example, 2 is subtracted from 4, yielding the solution 2. Remember to perform subtraction in the correct order, as reversing the minuend and subtrahend will change the result. Subtraction could also result in a negative number if the subtrahend is larger than the minuend, which is perfectly valid in many algebraic expressions.