Problem 48 - Elementary Algebra

Question

Solve \(p=m v\) for \(m\). Find the value of \(m\) when \(p=4240\) and \(v=260\).

Step-by-Step Solution

Answer

Question: Given that \(p=4240\) and \(v=260\), find the value of \(m\) in the equation \(p=mv\). Answer: The value of \(m\) is approximately \(16.3\).

Step 1: Solve the equation \(p=mv\) for \(m\)

To solve the equation for \(m\), we need to isolate \(m\) on one side of the equation. In this case, we'll divide both sides of the equation by \(v\): $$\frac{p}{v} = m$$

Step 2: Plug in the values of \(p\) and \(v\)

We are given \(p=4240\) and \(v=260\). Now, we can substitute these values into the equation to find the value of \(m\): $$m = \frac{4240}{260}$$

Step 3: Calculate the value of \(m\)

Now, we can perform the division to find \(m\): $$m = \frac{4240}{260} = 16.3$$ Therefore, the value of \(m\) is approximately \(16.3\).

Key Concepts

Algebra

Algebra is like the language of mathematics. It lets us describe mathematical relationships using symbols and letters. This makes it easier to work with numbers in a flexible way. In algebra, we use letters to represent numbers. These unknown numbers are called variables.
Algebra is important because it helps us find missing values in equations. For example, if the equation is given as \( p = mv \), it contains a relationship between three variables: \( p \) (momentum), \( m \) (mass), and \( v \) (velocity). Algebra lets us rearrange and solve such equations to find unknown values.

• Algebraic expressions consist of numbers, variables, and operations.
• Equations are statements indicating that two algebraic expressions are equal.

This understanding helps us solve practical problems by finding the value of an unknown variable, as we did in the example equation \(p = mv\).

Isolation of Variables

Isolation of variables means getting a variable by itself on one side of an equation. This is a fundamental step when solving for a specific variable. It allows us to clearly understand how other variables affect this one variable.
For example, to solve the equation \( p = mv \) for \( m \), we need to isolate \( m \). Here's how we can do it:

• Start with the original equation: \( p = mv \).
• We need \( m \) alone, so we divide both sides by \( v \): \( \frac{p}{v} = m \).

By isolating \( m \), we now have \( m \) on one side of the equation. This isolation helps to easily substitute known values into the equation to find the unknown value of \( m \). This technique is useful because it can be used for any variable in an equation.

Substitution Method

The substitution method is a powerful tool that involves replacing a variable with its known value. This method is often used after isolating the variable we are interested in. It simplifies equations so that they can be solved directly.
In the exercise, once we isolated \( m \) to get \( m = \frac{p}{v} \), we substituted the given values \( p = 4240 \) and \( v = 260 \):

• Substitute these values into the equation \( m = \frac{4240}{260} \).
• This gives us the simple arithmetic calculation: \( m = 16.3 \).

The substitution method is useful not only in solving equations but also in making complicated expressions simpler. It provides an uncomplicated path to reach a solution by using known values to find unknowns efficiently.