Problem 11 - Algebra Form and Function

Question

Solve the equation \(g(t)=a\) given that: $$ g(t)=2(t-1)+4(2 t+3) \text { and } a=0 $$

Step-by-Step Solution

Answer

Answer: The value of t is -1 when g(t) = 0 for the given function.

Step 1: Distribute constants in the g(t) function

Distribute the constants within the parentheses in the given expression for g(t): $$ g(t) = 2(t - 1) + 4(2t + 3) $$

Step 2: Simplify the g(t) function

Combine like terms to simplify the g(t) function: $$ g(t) = 2t - 2 + 8t + 12 $$ $$ g(t) = 10t + 10 $$

Step 3: Set g(t) equal to the given value of a

We are given that a = 0. Substitute 0 for g(t) in the equation to set up the equation we need to solve for t: $$ 0 = 10t + 10 $$

Step 4: Solve for t

Subtract 10 from both sides and then divide by 10 to find the value of t: $$ -10 = 10t $$ $$ t = -1 $$ The solution to the equation g(t) = a is t = -1.

Key Concepts

Distributive Property

The distributive property is a crucial mathematical rule that helps simplify expressions and solve equations. It enables you to spread multiplication over addition or subtraction.When applying the distributive property, you multiply each term within a set of parentheses by a factor outside the parentheses.For the expression \( g(t) = 2(t - 1) + 4(2t + 3) \), the distributive property directs you to multiply:

• 2 by each term in \( (t - 1) \), resulting in \( 2t - 2 \).
• 4 by each term in \( (2t + 3) \), resulting in \( 8t + 12 \).

This property is valuable because it makes more complex expressions easier to handle, as it breaks them down into simpler terms.

Linear Function

Linear functions are algebraic equations that form a straight line when graphed on a coordinate plane. These functions are typically in the form \( f(x) = mx + b \), where \( m \) and \( b \) are constants.In our exercise, the function \( g(t) \) simplifies to \( 10t + 10 \), which is a linear equation.Here:

• \( m = 10 \), representing the slope or rate of change.
• \( b = 10 \), denoting the y-intercept, the point where the line crosses the y-axis.

Linear functions like this indicate a constant rate of change and can be easily solved for unknown variables. They simplify the process of finding solutions to equations by providing a clear, straightforward relationship between variables.

Simplification of Expressions

Simplification involves reducing expressions to their simplest form, making them easier to understand and work with. In the provided solution, terms in the expression \( g(t) = 10t + 10 \) have been combined:

• First, the expression \( 2t - 2 + 8t + 12 \) comes from distributing as mentioned before.
• Then, similar terms, \( 2t \) and \( 8t \), are added together to form \( 10t \).
• Likewise, constant terms \(-2 \) and \( +12 \) are combined to become \( +10 \).

Simplification is a powerful tool as it transforms complicated equations into manageable forms. By boiling down expressions to their core components, students gain clarity and ease in further solving processes, such as setting the equation equal to a given value and isolating variables.