Problem 17 - Algebra Form and Function

Question

Rewrite the function in slope-intercept form. $$ g(n)=14-2 / 3(n-12) $$

Step-by-Step Solution

Answer

Question: Rewrite the given function in slope-intercept form: $$g(n) = 14 - \frac{2}{3}(n-12)$$ Answer: $$g(n) = -\frac{2}{3}(n) + 22$$

Step 1: Distribute the term with the variable

Utilize the distributive property to distribute the $$-2/3$$ to both terms inside the parentheses, resulting in: $$ g(n) = 14 - \frac{2}{3}(n) + \frac{2}{3}(12) $$

Step 2: Multiply the term with the constant value

Multiply $$\frac{2}{3}$$ by 12 to simplify the constant term: $$ g(n) = 14 - \frac{2}{3}(n) + 8 $$

Step 3: Combine constant terms

Combine the 14 and 8 to a single constant term: $$ g(n) = -\frac{2}{3}(n) + 22 $$ The function is now in slope-intercept form, with a slope of $$m=-\frac{2}{3}$$ and a y-intercept of $$b=22$$.

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle that allows us to simplify expressions. It states that multiplying a single term by terms inside parentheses, like \( a(b + c) \), is equivalent to \( ab + ac \). In other words, distribute the term outside the parenthesis to each term inside.
This is helpful when simplifying expressions with variables or constants.
In the original exercise, the term \( -\frac{2}{3} \) is distributed to both \( n \) and 12 inside the parentheses.
This results in:

• \( -\frac{2}{3}(n) \)
• \( -\frac{2}{3} \times 12 \)

Distributing terms like this can simplify and subsequently manipulate algebraic expressions more easily.

Simplifying Expressions

Once we apply the distributive property, the next step is to simplify the expression further. This involves performing any multiplication or division needed to reduce the terms to their simplest form.
In the solution provided, this means calculating \( -\frac{2}{3} \times 12 \) to simplify it to \(-8\).
This changes the equation to:

• \( g(n) = 14 - \frac{2}{3}n + 8 \)

The simplification process helps in reducing complex expressions to a form that is easier to work with, eventually making it simpler to interpret or graph the function when needed.

Combining Like Terms

Combining like terms is another essential technique in algebra that helps in simplifying an expression. This involves adding or subtracting terms that have the same variable part. It’s important for consolidating constants within an expression.
In our exercise, combining the constants 14 and 8 results in:

• \( 14 + 8 = 22 \)

With this process, our equation is simplified to \( g(n) = -\frac{2}{3}n + 22 \), clearly presenting the function in slope-intercept form. This form, \( y = mx + b \), shows the slope \( m \) as \(-\frac{2}{3}\) and the y-intercept \( b \) as 22, making it straightforward for graphing purposes.