Problem 73 - Algebra and Trigonometry with Analytic Geometry

Question

The formula occurs in the indicated application. Solve for the specified variable. \(S=\frac{p}{q+p(1-q)}\) for \(q\)

Step-by-Step Solution

Answer

The solution for \(q\) is \( q = \frac{p - Sp}{S - Sp} \).

Step 1: Clear the Fraction

Multiply both sides of the equation by \(q + p(1-q)\) to eliminate the fraction. This gives us: \( S(q + p(1-q)) = p \).

Step 2: Distribute S

Distribute \(S\) throughout \(q + p(1-q)\):\[Sq + Sp(1-q) = p\]

Step 3: Simplify the Expression

Simplify the expression from the distribution step:\[ Sq + Sp - Spq = p \]

Step 4: Group Terms with q

Move all terms containing \(q\) to one side and constants to the other:\[ Sq - Spq = p - Sp \]

Step 5: Factor Out q

Factor \(q\) out of the left side of the equation:\[ q(S - Sp) = p - Sp \]

Step 6: Solve for q

Divide both sides of the equation by \((S - Sp)\) to solve for \(q\):\[ q = \frac{p - Sp}{S - Sp} \]

Key Concepts

Solving for a Variable

When you are tasked with solving an equation for a specific variable, the main goal is to isolate that variable on one side of the equation. This means you are rearranging the equation so that this variable stands alone on one side, while all other terms are moved to the opposite side. In algebra, this process often involves reversing operations that affect the variable, such as addition, subtraction, multiplication, or division. For instance, if a variable is being multiplied by a term, you would typically divide both sides by that term to undo the operation and isolate the variable.

Let’s look at an example based on the original exercise. We start with an equation like:

• Begin with the equation: \( S(q + p(1-q)) = p \)
• Our target is to solve for \( q \).

By methodically rearranging and simplifying, we can isolate \( q \). This process involves carefully undoing the operations around \( q \) by applying inverse operations. This is a critical skill in algebra, enabling us to analyze and understand mathematical relationships.

Equation Simplification

Equation simplification is all about making an equation easier to work with. This involves reducing complexity by getting rid of unnecessary terms and combining like terms. It’s like cleaning up a messy room, where the goal is to make everything neat and orderly. Simplifying equations also makes it easier to see the relationships between variables and constants, which is crucial in solving equations.

In our context, simplification played a crucial role after the fraction was cleared:

• After multiplying and expanding: \( Sq + Sp(1-q) = p \)
• Simplified to: \( Sq + Sp - Spq = p \)

The simplification makes it easier to factor and eventually solve for the variable of interest. By combining and canceling terms, you end up with a simpler expression that is much easier to work with. Remember, the keywords are reduce, combine, and clarify - these are your guiding principles when simplifying any equation.

Fraction Elimination

Fractions can complicate equations, making them harder to solve. Therefore, eliminating fractions is often the first step in simplifying and solving equations. To get rid of a fraction, multiply every term of the equation by the denominator of the fraction. This operation clears the fraction, turning it into a simpler format.

In the sample exercise, the fraction \( \frac{p}{q+p(1-q)} \) was eliminated in the very first step:

• Multiply each side by \( q + p(1-q) \) to eliminate the fraction: \( S(q + p(1-q)) = p \)

Notice how this action simplifies the equation and makes it easier to handle in subsequent steps. This critical step can simplify many algebra problems, making it an essential tool in your algebraic toolbox. By understanding and applying fraction elimination, you will find algebra much more approachable and less intimidating.