Problem 63 - Elementary Algebra

Question

For the following problems, solve the literal equations for the indicated variable. When directed, find the value of that variable for the given values of the other variables. Solve \(t=\frac{Q+6 P}{8}\) for \(P\).

Step-by-Step Solution

Answer

Question: Solve the given equation for \(P\): \(t=\frac{Q+6 P}{8}\) Answer: \(P=\frac{8t-Q}{6}\)

Step 1: Multiply both sides by 8

Since we want to isolate the term containing \(P\) in the numerator, let's start by multiplying both sides of the equation by 8 to eliminate the denominator: \(8\cdot t = 8 \cdot \frac{Q+6 P}{8}\) This simplifies to: \(8t=Q+6P\)

Step 2: Subtract \(Q\) from both sides

In order to isolate \(P\), we need to remove the term \(Q\) from the equation. We can do that by subtracting \(Q\) from both sides: \(8t - Q = Q + 6P - Q\) This simplifies to: \(8t - Q = 6P\)

Step 3: Divide by 6

Finally, to isolate \(P\) completely, we need to divide both sides of the equation by 6: \(\frac{8t-Q}{6}=\frac{6P}{6}\) This simplifies to: \(\frac{8t-Q}{6}=P\) Now we have successfully solved the equation for the variable \(P\). The final equation in terms of \(P\) is: \(P=\frac{8t-Q}{6}\)

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a fundamental skill when it comes to understanding and solving equations. It involves various operations such as adding, subtracting, multiplying, and dividing terms to rearrange and simplify equations. Mastery of algebraic manipulation is essential to rearrange equations and solve for unknown variables.

For example, in solving the equation given in the exercise, multiple steps of algebraic manipulation are necessary. We begin by multiplying both sides of the equation by 8 to clear the fraction. The equation transforms from a ratio to a linear equation, where further manipulations are straightforward. Understanding the properties of equality and the distributive property allows us to perform operations that keep the equation balanced. Algebraic manipulation provides a structured approach to solving complex problems by breaking them down into simpler, more manageable steps.

Isolating Variables

Isolating variables is a key objective in solving literal equations. The aim is to rewrite the equation so that the variable of interest is on one side by itself. This typically involves a series of algebraic manipulations.

Using the example from the textbook exercise, isolating the variable, in this case, means getting the letter P on one side of the equation. To achieve this, we move all other terms to the opposite side. In step 2, we subtract Q from both sides, reducing the equation to an expression with only terms containing P on one side. Isolating the variable often requires these types of operations, performed with care to ensure that the balance and meaning of the equation are maintained.

Literal Equation Solutions

Literal equation solutions refer to equations where the focus is on variables rather than specific numbers. In the classroom problem, we encounter a literal equation that expresses t in terms of P and Q. The process to solve a literal equation is akin to solving numerical equations, wherein one finds the value of one variable in terms of others.

In solving for P, we do not end up with a numeric answer but rather an expression that relates P to t and Q. Understanding how different variables relate to one another is critical in various fields such as physics, economics, and engineering, where equations often involve multiple variables and constants that describe relationships rather than single outcomes.

Equation Transformations

Equation transformations involve changing the form of an equation while keeping its solutions equivalent. This can be seen as a powerful tool that simplifies complex relationships into more digestible forms. The transformations follow algebraic principles and aim to modify the equation to better suit the problem at hand.

In the textbook example, transforming the original equation begins with multiplying by 8 to eliminate the fraction. Again, subtracting Q from both sides transforms it further, bringing us closer to the variable of interest. Lastly, dividing by 6 results in the isolate form of P. Transforming equations is a versatile technique and is especially useful when dealing with equations involving several variables, as it allows us to gain insights into how changing one variable affects others.