Problem 68 - Elementary and Intermediate Algebra

Question

Suppose \(h=4 n\) and \(n=2 g .\) Complete the following formula: h= ___ g

Step-by-Step Solution

Answer

The complete formula is \(h = 8g\).

Step 1: Understand the Given Expressions

We are provided with the expressions \(h = 4n\) and \(n = 2g\). Our goal is to express \(h\) in terms of \(g\).

Step 2: Substitute Expression for n in h

Since \(n = 2g\), we can substitute \(n\) with \(2g\) in the expression for \(h\):\[h = 4n = 4(2g)\].

Step 3: Simplify the Expression

Now simplify the expression \(4(2g)\):\[h = 4 imes 2 imes g = 8g\]. This results in the formula \(h = 8g\).

Key Concepts

Substitution Method

The substitution method is a powerful technique used to simplify algebraic expressions by replacing one variable with another, usually based on a given relationship between the two variables. In this exercise, we have the equations:

• \(h = 4n\)
• \(n = 2g\)

Our aim is to express \(h\) in terms of \(g\). To achieve this, we replace the variable \(n\) in the first equation with its equivalent expression in terms of \(g\) from the second equation, i.e., \(n = 2g\). This way, we substitute \(n\) with \(2g\), transforming the equation for \(h\) into: \[h = 4n = 4(2g)\] This step significantly simplifies the direct relationship where \(h\) depends on \(g\) by using known relationships between the variables.

Simplifying Expressions

Simplifying expressions is a critical step in algebra to make equations easier to understand and work with. Once we've substituted \(n\) with \(2g\) into the equation for \(h\), we need to simplify the expression \(4(2g)\).This involves performing the arithmetic operation within the expression:- Multiply the coefficients: \(4 \times 2 = 8\)- Attach the variable \(g\) to the result of the multiplication, making the expression \(8g\)Therefore, the original expression \(4(2g)\) simplifies to \(8g\). This simplification helps in understanding the proportional relationship between \(h\) and \(g\), resulting in the equation \(h = 8g\). The process of simplifying expressions by combining like terms or performing arithmetic operations ensures clarity and accuracy in solving equations.

Proportional Relationships

Proportional relationships describe how one variable changes in relation to another in a consistent way. In this exercise, the ultimate expression we derive is \(h = 8g\), which shows a proportional relationship between the variables \(h\) and \(g\). This relationship can be interpreted as \(h\) being eight times the value of \(g\). Here are some characteristics of proportional relationships:- **Constant Ratio:** The ratio of \(h\) to \(g\) is constant at 8:1.- **Linear Representation:** If graphed, it creates a straight line passing through the origin.This equation indicates that for every increase in \(g\), the value of \(h\) increases by eight times the change in \(g\). It illustrates how changes in one variable are directly reflected in another in a predictable and linear manner. Understanding proportionality is fundamental in solving many algebraic problems, as it helps establish the relationship between variables in a simple form.