Half of a number decreased by 8 is – 3 – Half of a number decreased by 8 is -3 presents a mathematical conundrum that invites exploration. This equation, seemingly simple at first glance, unveils a rich tapestry of algebraic concepts and problem-solving techniques. As we delve into its intricacies, we will uncover the underlying principles that govern this equation and its applications in the realm of word problems.

To embark on this mathematical journey, we will first dissect the equation into its constituent parts, examining each term and its relationship to the unknown number. We will then introduce a variable to represent the unknown, enabling us to express the equation in a more concise and versatile form.

With the equation suitably transformed, we will proceed to solve it step-by-step, employing fundamental algebraic operations and logical reasoning.

Half of a Number Decreased by 8 is

3

The equation “half of a number decreased by 8 is -3” can be expressed mathematically as:

$\frac12x

• 8 =
• 3$

where $x$ represents the unknown number.

Variable Representation

The variable $x$ represents the unknown number that we are trying to find. In the equation, $x$ is being multiplied by $\frac12$ to find half of the number, and then 8 is being subtracted from the result.

Solving the Equation

To solve the equation, we need to isolate the variable $x$ on one side of the equation and the constant on the other side.

1. Add 8 to both sides of the equation:

$\frac12x

• 8 + 8 =
• 3 + 8$

2. Simplify:

$\frac12x = 5$

3. Multiply both sides of the equation by 2 to get rid of the fraction:

$2 \cdot \frac12x = 2 \cdot 5$

4. Simplify:

$x = 10$

Numerical Solution

Therefore, the unknown number is $x = 10$. We can verify this by substituting this value back into the original equation:

$\frac12(10)

• 8 =
• 3$

$5

• 8 =
• 3$

$-3 =

3$

Since both sides of the equation are equal, we have verified that $x = 10$ is the correct solution.

Word Problem Application, Half of a number decreased by 8 is – 3

A farmer has a rectangular field with a length that is 8 meters more than twice its width. If the area of the field is 120 square meters, what is the width of the field?

Let $w$ be the width of the field. Then the length of the field is $2w + 8$. The area of the field is given by $A = lw$. Substituting these values into the area equation, we get:

$120 = w(2w + 8)$

Expanding and simplifying, we get:

$120 = 2w^2 + 8w$

$2w^2 + 8w

120 = 0$

Solving this quadratic equation using the quadratic formula, we get:

$w = 6$ or $w =

10$

Since the width of the field cannot be negative, we discard the solution $w = -10$. Therefore, the width of the field is $w = 6$ meters.

Question Bank: Half Of A Number Decreased By 8 Is – 3

What is the significance of the variable in the equation?

The variable serves as a placeholder for the unknown number, allowing us to represent the equation in a generalized form. It facilitates algebraic manipulation and enables us to find the numerical value of the unknown.

How can I apply this equation to solve word problems?

To apply this equation to word problems, identify the unknown quantity and translate it into a mathematical expression. Then, set up an equation based on the given information and solve for the unknown using the steps Artikeld in this exploration.