Problem 3 - Algebra for College Students

Question

Solve each equation for \(y\). $$M=\frac{1}{y}$$

Step-by-Step Solution

Answer

y = \frac{1}{M}

Step 1 - Understand the Equation

The given equation is \( M = \frac{1}{y} \). We need to solve for \( y \).

Step 2 - Isolate \( y \)

To isolate \( y \), we need to clear the fraction. Multiply both sides of the equation by \( y \) to get rid of the denominator: \( M \times y = 1 \).

Step 3 - Solve for \( y \)

Now, solve for \( y \) by dividing both sides by \( M \): \( y = \frac{1}{M} \).

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. In algebra, symbols, often letters, represent numbers or other elements. These symbols and their operations follow specific rules, enabling us to solve equations like the one in this exercise.
For instance, in the equation given, variables such as M and y are used. This allows us to express relationships and solve for unknown values. Understanding algebra is crucial as it forms the foundation for higher mathematics. It also helps in various real-life applications, such as in engineering and science.

Isolating Variables

Isolating variables is a key step in solving equations. It involves manipulating the equation to get the variable of interest by itself on one side of the equation.
In the provided exercise, the goal is to solve for the variable y. The first step is to clear the fraction by multiplying both sides of the equation by y, resulting in: \[ M y = 1 \]
Next, to isolate y, divide both sides by M:\[ y = \frac{1}{M} \]
This process ensures y is alone on one side, giving the solution in terms of M. Isolating variables is essential in algebra as it simplifies equations and helps find solutions.

Fractions

Fractions represent parts of a whole and consist of a numerator (top part) and a denominator (bottom part). In equations, fractions can sometimes complicate the process of solving for a variable.
In the exercise, the fraction \[ \frac{1}{y} \] represents a relationship between M and y. To solve for y, we need to eliminate the fraction. This is done by multiplying both sides by y, converting the fraction into a simpler form: \[ M y = 1 \]
After this, solving for y involves straightforward division. Understanding how to handle fractions is fundamental in algebra, as it allows for the simplification and solving of various types of equations.