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Chapter 2: Problem 99

Combine like terms. [ 1.3] Solve for \(m: x=\frac{m p}{c}\).

Short Answer

Expert verified
The solution is: \( m = \frac{cx}{p} \)

Step by step solution

01

- Understand the equation

The given equation is: \[ x = \frac{mp}{c} \]
02

- Isolate the variable m

To isolate the variable \(m\), we need to remove the fraction. Multiply both sides of the equation by \(c\): \[ c \cdot x = c \cdot \frac{mp}{c} \]
03

- Simplify the equation

Simplify the right side of the equation. The \(c\) terms cancel out, leaving: \[ cx = mp \]
04

- Solve for m

To solve for \(m\), divide both sides of the equation by \(p\): \[ \frac{cx}{p} = m \]
05

- Final answer

Rearrange the equation to show the solution: \[ m = \frac{cx}{p} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolate the Variable
Isolating a variable means solving an equation to get that variable alone on one side. This is often necessary when we want to find the specific value of the variable. In our exercise, we need to isolate the variable \( m \) from the equation \( x = \frac{mp}{c} \). Here's how to go about it:

First, identify the variable you want to isolate. In this case, it's \( m \).
Next, use algebraic operations to move all other terms to the opposite side of the equation. Think of it like peeling off layers until \( m \) is all by itself. For example, we have a fraction involving \( m \), so we'll work on removing that fraction.
Remove Fractions
Fractions can complicate algebraic manipulation, so it's often helpful to remove them early. In this exercise, our main goal is to clear the fraction on the right side of the equation \( x = \frac{mp}{c} \).

To remove the fraction, we should multiply both sides of the equation by the denominator of the fraction, which is \( c \). This operation neutralizes the denominator:
\[ c \times x = c \times \frac{mp}{c} \]
On the right-hand side, the \( c \) in the numerator and the \( c \) in the denominator cancel each other out. This leaves us with:

\[ cx = mp \]
Now, we have a simplified equation without fractions, which makes it easier to solve for \( m \).
Algebraic Manipulation
Algebraic manipulation involves using various operations to simplify equations and solve for variables. After removing the fraction in our exercise, we have:\( cx = mp \)

Next, to isolate \( m \), we need to divide both sides of the equation by \( p \). This helps balance the equation and keeps \( m \) by itself:
\[ \frac{cx}{p} = m \]

Finally, rearrange this to show \( m \) on the left side, making it easier to read:
\[ m = \frac{cx}{p} \]
So, we've isolated \( m \) using algebraic manipulation, starting with multiplication to clear the fraction and finishing with division. Mastering these manipulations is key to efficiently solving algebraic equations.

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