Problem 12 - Introductory Algebra for College Students

Question

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(P=C+M C\) for \(M\)

Step-by-Step Solution

Answer

The solution for \(M\) in the formula \(P=C+M C\) is \(M = \frac{(P - C)}{C}\).

Step 1: Identify the Variable

Recognize which variable the problem asks to solve for, in this instance, \(M\) is the specified variable.

Step 2: Isolate the Term Containing the Specified Variable

To isolate \(M\), begin by subtracting \(C\) from both sides of the equation. This results in \(P - C = MC\). This operation was performed to remove \(C\) from the left side of the equation.

Step 3: Solve for the Specified Variable

Finally, to solve for \(M\), divide both sides of the equation by \(C\). This gives \(M = \frac{(P - C)}{C}\). The operation is performed to get \(M\) by itself on one side of the equation.

Key Concepts

Solving Equations

When you're faced with an equation like \(P = C + MC\), the primary goal is to find the value of a specific variable. In this exercise, that variable is \(M\). Most equations are balanced statements—they indicate that two expressions are equal. Solving an equation means finding the value of the variable that makes this statement true.
To solve the equation, we must perform operations that transform it step by step, keeping it balanced at all times.
Simply put:

• Start by identifying what you need to solve for—here, it's \(M\).
• Apply arithmetic operations to both sides to maintain equality.
• Ensure each operation moves you closer to isolating the variable.

This logical approach helps in systematically uncovering the value of the variable required.

Isolating Variables

Isolating a variable is crucial in algebra. This means you want the variable to stand alone on one side of the equation. For \(P = C + MC\), isolating \(M\) requires removing other terms that share its side.
Here's the strategic plan shown step by step:

• Remove constant terms: Subtract \(C\) from both sides. You get \(P - C = MC\). This operation takes away unnecessary terms to clear the path towards \(M\).
• Separate the variable from its factor: Divide each side by \(C\). This yields \(M = \frac{(P - C)}{C}\), beautifully presenting \(M\) alone.

Isolating rightly is akin to putting the variable in the spotlight, removing the clutter that hides it.

Variables in Equations

Variables are symbols used to represent unknowns in mathematical expressions or equations. In our formula \(P = C + MC\), the variables are \(P\), \(C\), and \(M\). Understanding their role is important as they provide flexibility and the ability to solve different problems with a single equation.
Here’s what to know about the variables in an equation:

• Symbols of Change: Variables like \(P\), \(C\), and \(M\) often stand for quantities that can vary or change.
• Substitution: Once solved, they can be replaced with numerical values to represent real-world situations.
• Interrelation: In an equation, variables typically relate to one another, and changes in one can affect others.

Grasping the concept of variables helps build a strong foundation for solving equations efficiently.