In the realm of algebra, we encounter intriguing problems that challenge our understanding of numerical relationships. One such puzzle is “half of a number decreased by 8 is 3.” This mathematical conundrum invites us to embark on a journey of variable representation, equation solving, and result interpretation, revealing the hidden logic behind the statement.
Our exploration begins by representing the unknown number as a variable, x. Translating the given statement into mathematical terms, we arrive at the equation: (1/2)x – 8 = 3. This equation serves as the cornerstone of our quest to uncover the value of x.
Variable Representation
In order to solve this problem, we need to represent the unknown number as a variable. Let’s call the number x.
The statement “half of a number decreased by 8 is 3” can be translated into the following equation:
Equation
(x/2)
8 = 3
Solving for the Variable
Solving for the variable in an equation involves isolating it on one side of the equation while ensuring that the equation remains balanced. This process requires careful manipulation of the equation, following specific steps to simplify and isolate the variable.
Simplifying the Equation
To simplify the equation, we need to remove any unnecessary terms or coefficients that complicate the equation. This can involve combining like terms, removing parentheses, and simplifying fractions or decimals.
Example:Simplify the equation: 2x + 5 = 11Subtract 5 from both sides: 2x = 6Divide both sides by 2: x = 3
Isolating the Variable
Once the equation is simplified, we can isolate the variable by moving all the terms containing the variable to one side of the equation and all the constant terms to the other side.
- Addition or Subtraction:If the variable is added or subtracted from a constant, we can isolate it by performing the inverse operation on both sides of the equation.
- Multiplication or Division:If the variable is multiplied or divided by a constant, we can isolate it by performing the inverse operation on both sides of the equation.
Example:Solve for x: 3(x
2) = 15
Distribute the 3: 3x
6 = 15
Add 6 to both sides: 3x = 21Divide both sides by 3: x = 7
Interpreting the Result
The value of x obtained in the previous step is of paramount importance in understanding the original statement and its implications.
The result, x = 16, represents the number that satisfies the condition given in the statement. When half of this number is decreased by 8, the result is 3.
Significance of the Result
The significance of the result lies in its ability to provide a concrete numerical value for the unknown number in the statement. By solving for x, we have determined the specific number that fulfills the given condition.
Moreover, the result allows us to draw inferences about the original statement and its context. It enables us to understand the relationship between the unknown number and the other values mentioned in the statement.
Illustrating with an Example
To solidify our understanding of the steps involved in solving for the variable in the given statement, let’s work through a specific numerical example.
Suppose we are given the statement: “Half of a number decreased by 8 is 3.” We can represent this statement mathematically as:
$\frac12x
8 = 3$
Where x represents the unknown number.
Applying the Steps, Half of a number decreased by 8 is 3
- Isolate the variable term:Add 8 to both sides of the equation to isolate the variable term on one side:
- $\frac12x – 8 + 8 = 3 + 8$
- $\frac12x = 11$
- Solve for x:Multiply both sides of the equation by 2 to solve for x:
- $2 \cdot \frac12x = 2 \cdot 11$
- $x = 22$
Therefore, the number that satisfies the given statement is 22.
Question Bank: Half Of A Number Decreased By 8 Is 3
What is the first step in solving the equation “half of a number decreased by 8 is 3”?
The first step is to represent the unknown number as a variable, x, and translate the statement into the equation: (1/2)x – 8 = 3.
How do we solve for the variable x in the equation?
To solve for x, we isolate the variable on one side of the equation by performing algebraic operations such as adding 8 to both sides and multiplying both sides by 2.
What does the result of solving for x represent?
The result of solving for x represents the value of the unknown number that satisfies the original statement “half of a number decreased by 8 is 3.”
