Problem 27 - Algebra Form and Function

Question

Put the functions in the form \(Q=k t\) and state the value of \(k\). $$ Q=b t+r t $$

Step-by-Step Solution

Answer

Answer: \(Q = kt\) where \(k = b + r\).

Step 1: Identify the common factor

The common factor in the given equation \(Q = bt + rt\) is \(t\) (the variable representing time).

Step 2: Factor out the common factor

Since the common factor is \(t\), we can factor it out from the given equation. Doing this, we get: \(Q = t(b + r)\)

Step 3: Rewrite the function in the form \(Q=kt\)

Now that we have factored out \(t\), we can rewrite the function in the form \(Q = kt\). In our case, the constant \(k\) is the sum of \(b\) and \(r\): \(Q = t(b + r)\) is equivalent to \(Q = kt\) where \(k = (b + r)\).

Step 4: State the value of k

The value of \(k\) is the sum of \(b\) and \(r\). So, \(k = b + r\).

Key Concepts

Linear Functions

A linear function is an algebraic expression that, when graphed, forms a straight line. It is one of the simplest and most common types of functions used in mathematics. These functions can be defined by an equation of the form \( Q = kt \), where \( Q \) and \( t \) are variables, and \( k \) is a constant. Linear functions describe proportional relationships between their variables. When you plug different values for \( t \) into the equation, the corresponding \( Q \) values change in a predictable, steady way.

Linear functions are essential in understanding relationships in real-world situations, such as speed (distance over time) or economics (cost over quantity). In the given exercise, the function \( Q = bt + rt \) is translated into a linear form \( Q = kt \) by factoring, which makes it easier to interpret the relationship between the variables involved.

Common Factors

The concept of a common factor is crucial in both arithmetic and algebra. A common factor is a number or algebraic term that divides two or more numbers or expressions evenly without leaving a remainder. Identifying common factors helps simplify algebraic expressions and solve equations more efficiently.

For example, consider the expression \( Q = bt + rt \). Both terms in the equation contain \( t \), making it a common factor. Factoring out the common factor \( t \) results in \( Q = t(b + r) \). This significantly simplifies the expression and helps us represent it in different useful forms, like the linear function format \( Q = kt \).

Understanding how to find and factor out common factors will improve your ability to work with more complex mathematical problems and algebraic manipulations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (such as addition, subtraction, multiplication, and division). They are essential in forming the basis of algebraic reasoning and problem-solving. These expressions can represent real-world quantities or abstract mathematical concepts.

The exercise provided involves manipulating the algebraic expression \( Q = bt + rt \). By recognizing that \( t \) is a common factor, the problem becomes more manageable; allowing us to simplify it to \( Q = t(b + r) \). This simplification transforms a potentially complex expression into a more understandable form.

Working with algebraic expressions often involves rewriting them, factoring, and applying operations. These skills are vital for solving equations, understanding functions, and modeling real-world situations effectively. Recognizing different forms and structures within algebraic expressions lets students solve problems with greater ease and accuracy.