Problem 31 - Algebra Form and Function

Question

Solve for the indicated variable. Assume all constants are non-zero. $$ 3 x t+1=2 t-5 x, \text { for } t, \text { if } x \neq 2 / 3 $$

Step-by-Step Solution

Answer

Question: Solve the equation \(3xt + 1 = 2t - 5x\) for \(t\), where \(x \neq 2/3\). Answer: \(t = \frac{-5x - 1}{3x - 2}\)

Step 1: Write the given equation

The given equation is: $$ 3 x t + 1 = 2 t - 5 x $$ We need to solve it for \(t\).

Step 2: Move terms with 't' to one side and terms without 't' to the other side

In order to isolate \(t\), we will move \(2t\) and \(1\) to opposite sides of the equation by subtracting \(2t\) from both sides and subtracting \(1\) from both sides: $$ 3xt - 2t = -5x - 1 $$

Step 3: Factor out 't'

Now, we can factor \(t\) out of the left side of the equation: $$ t(3x - 2) = -5x - 1 $$

Step 4: Divide both sides by the factor of 't'

To isolate \(t\), we will divide both sides of the equation by \((3x - 2)\): $$ t = \frac{-5x - 1}{3x - 2} $$ This is the final expression for \(t\) in terms of \(x\). Note that we are given the condition that \(x \neq 2/3\), which means the denominator of the above fraction will never be zero and the expression is valid.

Key Concepts

Isolation of Variables

In algebra, solving equations often requires the process known as the isolation of variables. This involves rearranging the equation to get the variable of interest on one side of the equation, all by itself. This step is vital for finding the solution in terms of other variables or constants.

To effectively isolate a variable, follow these steps:

• First, identify all terms that contain the desired variable.
• Move all terms containing the variable to one side of the equation and all other terms to the opposite side. The goal is to have the variable on one side and numbers or different variables on the other.
• Perform any necessary operations such as addition, subtraction, multiplication, or division to simplify the equation further.

In our original problem, the goal was to isolate the variable \(t\). By moving the terms involving \(t\) to one side, we could then factor \(t\) out, eventually isolating it, which led us to find its expression in terms of \(x\). Ensuring that all operations maintain equality on both sides is crucial to isolating the variable successfully.

Factorization

Factorization is a technique used to express an equation or expression as a product of its factors. This is particularly useful in the process of solving equations where multiple terms involve the same variable.

Here's how factorization works in solving equations:

• Start by identifying common factors in the terms of the equation.
• Extract these common factors so that a simpler equation is formed, which makes further solving more straightforward.
• Use the factored expression to perform other algebraic operations, such as division, to solve for your variable.

In the example we examined, after moving terms involving \(t\) to one side, we then factored \(t\) out. This step simplified the equation to the form \(t(3x - 2) = -5x - 1\), setting up the final step of dividing by the coefficient of \(t\) to complete the process of isolating it. Factorization not only simplifies the equation but also reveals relationships between terms that might not be immediately apparent.

Linear Equations

Linear equations are equations of the first degree, meaning they have variables raised only to the power of one. Solving linear equations is fundamental in algebra because they form the basis for most problems.

Characteristics of linear equations include:

• They typically appear in the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.
• The graph of a linear equation is a straight line.
• Solutions to linear equations often involve simple arithmetic operations.

In our problem, though the presentation involves two variables, \(x\) and \(t\), the manipulation to solve for \(t\) exhibits methods typical for linear equations. By isolating \(t\) and rearranging terms into a linear form, the solution \(t = \frac{-5x - 1}{3x - 2}\) was derived efficiently. Understanding linear equations provides a strong foundation for handling more complex algebraic tasks.