Problem 2 - Elementary Algebra

Question

\(p=\frac{10,000}{v} . \quad\) Determine the value of \(p\) if \(v=250\).

Step-by-Step Solution

Answer

Answer: When \(v=250\), the value of \(p\) is 40.

Step 1: Substitute the given value of \(v\) into the expression

First, we will substitute the value of \(v=250\) into the given expression: \(p=\frac{10,000}{v}\). So, we have: \(p=\frac{10,000}{250}\).

Step 2: Solve the expression for \(p\)

Now, we will divide 10,000 by 250 to find the value of \(p\). We get: \(p=\frac{10,000}{250}=40\). So, when \(v=250\), the value of \(p\) is 40.

Key Concepts

Equation Solving

Equation solving in algebra is like solving a puzzle. You have to find the unknown value that makes the equation true. In our exercise, the equation is given as \(p=\frac{10,000}{v}\). The task is to determine what \(p\) equals when we know the value of \(v\).

When you solve an equation, you need to focus on isolating the variable you are trying to find. Here, the variable is \(p\). The equation has already expressed \(p\) in terms of another variable, \(v\). This means we just need to substitute the given value to solve it! Understanding this can make equation solving much simpler and help solve many similar problems calmly and correctly.

Equation solving can involve various operations like addition, subtraction, multiplication, and here specifically, division. Calmly working through each step ensures that you accurately find the value you are looking for.

Substitution Method

The substitution method is a key concept in algebra that simplifies solving equations. When you see an equation such as \(p=\frac{10,000}{v}\) and you know one of the variable's value, you can plug it directly into the equation. This is exactly what we did in the exercise by substituting \(v=250\). This changes the equation to \(p=\frac{10,000}{250}\).

By substituting the known value, you transform the algebraic expression into a simple arithmetic problem, making it far easier to handle. Once substituted, the expression to solve is straightforward. Substitution is particularly useful in equations involving two variables, making one variable's value dependent on the other's value.

• Identify the variable to substitute.
• Replace the variable in the equation with its given value.
• Simplify or solve the resulting equation.

This method gives life to numbers, turning abstract algebra into easy-to-manage arithmetic.

Division in Algebra

Division in algebra is a fundamental operation necessary for solving equations, especially when expressing one variable as a fraction involving another variable. Our example equation is \(p = \frac{10,000}{v}\), where we needed to divide 10,000 by a known \(v\) to find \(p\).

Understanding how to handle division in algebra helps in manipulating and solving a large range of equations. In algebraic expressions, division separates a total into equal parts based on the divisor. It's pretty much like fair sharing among friends! Here's how you'd approach dividing in an algebraic context:

• Recognize the parts: numerator (the number on top, 10,000) and denominator (the number on the bottom, \(v\)).
• Divide the numerator by the denominator.
• This gives the quotient, or the solution for the variable you're solving for, in this case, \(p\).

Once you grasp division in algebra, you'll be able to dissect complicated expressions down into understandable segments. In our exercise, dividing 10,000 by 250 was the key to finally finding that \(p = 40\).

Division, as one of four primary operations in arithmetic, bridges the gap from simple arithmetic to more complex algebraic operations.