Problem 49 - Elementary Algebra

Question

In the following problems, solve each of the conditional equations. Solve \(\frac{8 r s t}{3 p}=-2 p r s\) for \(t\)

Step-by-Step Solution

Answer

Question: Solve for 't' in the given equation: \(\frac{8 r s t}{3 p} = -2 p r s\) Answer: \(t = \frac{-3p^2}{4}\)

Step 1: Identify the variable to be solved for

In this case, we need to solve for the variable 't'.

Step 2: Eliminate common terms from both sides of the equation

We observe that the terms 'r' and 's' appear on both sides of the equation, so we can eliminate them: \(\frac{8 t}{3 p} = -2 p\)

Step 3: Isolate the variable 't' by multiplying both sides by '3p'

To obtain 't' on one side of the equation, we will multiply both sides of the equation by '3p': \(8t = -6p^2\)

Step 4: Solve for 't' by dividing both sides by '8'

Finally, to find the value of 't', we will divide both sides of the equation by '8': \(t = \frac{-6p^2}{8}\)

Step 5: Simplify the expression for 't'

We can simplify the expression by dividing both the numerator and the denominator by '2': \(t = \frac{-3p^2}{4}\) Now, we have the value of 't' in terms of the other variables: \(t = \frac{-3p^2}{4}\)

Key Concepts

Conditional Equations

A conditional equation is an equation that holds true only under certain conditions or for particular values of the variables involved. For instance, in the equation \( \frac{8 r s t}{3 p} = -2 p r s \), the equation is only true for certain values of \( t \), depending on the values of \( p \), \( r \), and \( s \).
Recognizing whether an equation is conditional is crucial because:

• The solutions depend on particular variable values, not universally true for any value.
• Understanding helps manage expectations of possible solutions.

To solve a conditional equation, we generally manipulate the equation to find the specific conditions or values that satisfy the equation.

Isolating Variables

Isolating a variable means rearranging an equation so that the variable stands alone on one side. This process involves performing the same mathematical operation on both sides of the equation. The goal is to have the variable of interest isolated so its value can be clearly determined from the equation.
For example, to solve for \( t \) in \( \frac{8 r s t}{3 p} = -2 p r s \), we first remove common terms 'r' and 's' to simplify and get \( \frac{8 t}{3 p} = -2 p \).
Then by multiplying both sides by \( 3p \), we can isolate \( t \):

• This gives us \( 8t = -6p^2 \).
• Finally, dividing both sides by '8', we achieve \( t = \frac{-6p^2}{8} \).

By isolating \( t \), we can find its particular value relative to other variables in the equation.

Simplifying Expressions

Simplifying an expression involves rewriting it in a more concise or efficient form without changing its value. It often involves reducing fractions, combining like terms, or using arithmetic operations to simplify numbers.
For example, after isolating \( t \), we found \( t = \frac{-6p^2}{8} \).
The expression can be simplified further:

• Both the numerator and denominator are divisible by 2.
• Dividing by 2 gives \( t = \frac{-3p^2}{4} \).

Simplifying helps in understanding the relationship between variables more easily and often reveals insights that are less obvious in the original complex form.