Problem 70 - Algebra Form and Function

Question

Put the equation \(y=3 x t+2 x t^{2}+5\) in the form \(y=b+m x .\) What are the values of \(b\) and \(m ?\) [Note: Your answers could include \(t\).]

Step-by-Step Solution

Answer

Question: Rewrite the equation y=3xt+2xt^2+5 in the form y=b+mx and identify the values of 'b' and 'm'. Answer: The given equation can be rewritten in the form y=b+mx as y = 5 + x(3t+2t^2), where b = 5 and m = (3t+2t^2).

Step 1: Identify the terms in the equation with x and without x

The given equation is y=3xt+2xt^2+5. In this equation, we have two terms with x (3xt, 2xt^2) and one term without x (5).

Step 2: Rewrite the terms with x as a single term with x

We know that 3xt and 2xt^2 are the terms with x in our equation. We need to rewrite these terms as a single term by factoring out x. So, we get x(3t+2t^2) as the x term.

Step 3: Rewrite the equation in the form y=b+mx

Now, we have the term with x (x(3t+2t^2)) and the term without x (5) in our equation. We can now rewrite the equation in the form y=b+mx: y = 5 + x(3t+2t^2) Here, b=5 and m=(3t+2t^2).

Step 4: Identify the values of b and m

In our final equation y = 5 + x(3t+2t^2), we can see that b = 5 and m = (3t+2t^2). So, these are the values of b and m.

Key Concepts

Factoring

Factoring is a mathematical process where you break down an expression into a product of simpler expressions. This is often used to simplify solutions or reveal underlying structures in equations. In the context of linear equations, factoring involves identifying and extracting common factors from terms.In the exercise, the equation is given as:\[ y = 3xt + 2xt^2 + 5. \]Here, both terms, \(3xt\) and \(2xt^2\), share a common factor, which is \(x\). This means we can factor \(x\) out of these terms to simplify the equation. By doing so, the expression becomes:\[ x(3t + 2t^2). \]Factoring is an important skill because it helps in simplifying equations and makes it easier to understand the structure of the equation. Keep in mind that factoring does not change the overall expression but makes it easier to work with by reducing it to simpler components. Learning to identify common factors and extract them efficiently is essential when dealing with more complex algebraic expressions.

Equation Rearrangement

Equation rearrangement is about restructuring an equation to isolate and highlight certain components or variables. This is often essential in linear equations when you need to compare or solve for dual variables. In this exercise, we restructured the original equation to conform to the form \(y = b + mx\).Starting with the given equation:\[ y = 3xt + 2xt^2 + 5, \]we took the step of factoring out \(x\) to get the expression \(x(3t + 2t^2)\). The equation was effectively simplified into a more familiar linear form:\[ y = 5 + x(3t + 2t^2), \]where the goal was to isolate the terms with \(x\) from the constant term. Rearranging simplifies recognizing that the term without \(x\) (5) becomes \(b\), and the factor multiplying \(x\) (\(3t + 2t^2\)) becomes \(m\).Rearranging equations is a useful technique in algebra that makes it easier to identify how each part of an equation contributes to the solution.

Variables and Constants

Understanding the difference between variables and constants is a cornerstone of algebra. They play significant roles in forming and interpreting equations. In any given equation, identifying these elements helps in predicting the equation's behavior and solutions.- **Variables**: These are symbols used to represent unknown or changing quantities. They can take different values. In the exercise, \(x\) and \(t\) are variables. They represent numbers that can change. In the context of the equation, they directly influence the value of \(y\).- **Constants**: These are fixed values that do not change within the context of an equation. In the rearranged equation \(y = 5 + x(3t + 2t^2)\), the number 5 is a constant, named as \(b\). It provides a referential fix in the equation that the solution pivots around.Recognizing variables and constants is crucial. Variables provide the flexibility for equations to describe more dynamic, complex situations, whereas constants ensure that certain conditions remain unchanged. In linear equations, the interplay between these components forms the basis of solving and understanding algebraic relationships.