Problem 34 - Elementary Algebra

Question

Solve the equations. $$ \frac{m}{-5}+4=-1 $$

Step-by-Step Solution

Answer

Question: Solve the linear equation for the variable \(m\): \(\frac{m}{-5} + 4 = -1\) Answer: \(m = 25\)

Step 1: Add constant terms to both sides

We're starting with the equation: $$ \frac{m}{-5} + 4 = -1 $$ Now, we'll subtract 4 from both sides of the equation to get rid of the constant on the left side. $$ \frac{m}{-5} + 4 - 4 = -1 - 4 $$

Step 2: Simplify the equation

After subtracting 4 from both sides, we have a simplified equation: $$ \frac{m}{-5} = -5 $$

Step 3: Isolate \(m\) by multiplying by reciprocal

Now we want to get \(m\) by itself by multiplying both sides of the equation by the reciprocal of the coefficient of \(m\). In our case, the coefficient is \(\frac{1}{-5}\) and its reciprocal is \(-5\). $$ -5 \times \frac{m}{-5} = -5 \times -5 $$

Step 4: Solve for \(m\)

After multiplying both sides with \(-5\), we have: $$ m = 25 $$ Now we have found the solution to the given equation, which is \(m = 25\).

Key Concepts

Algebraic Equations

Algebraic equations are foundational to understanding algebra and solving various real-world problems. An equation represents the balance between two expressions. In the given exercise, we tackle a linear equation, which is a type of algebraic equation where each term is either a constant or the product of a constant and a single variable.

For instance, the equation \(\frac{m}{-5} + 4 = -1\) features a variable \(m\), constants (4 and -1), and operations (division and addition). To solve this equation, the main goal is to determine the value of \(m\) that satisfies the balance. This process involves several steps, which include simplifying terms and isolating the variable to one side.

Isolating Variables

Isolating the variable is a fundamental step in solving algebraic equations and is the heart of finding a solution. It involves moving all occurrences of the variable to one side of the equation and all constants to the other side through the application of inverse operations.

For the exercise \(\frac{m}{-5} + 4 = -1\), we first aim to isolate the variable term \(\frac{m}{-5}\) by eliminating the constant +4. By subtracting 4 from both sides, we obtain \(\frac{m}{-5} = -5\). This technique of keeping the equation balanced while removing unwanted terms is not only crucial for linear equations but also sets the precedent for more complex equations involving multiple variables and operations.

Reciprocal Operations

Reciprocal operations are used to simplify equations and isolate variables. A reciprocal, also known as a multiplicative inverse, is a pair of numbers that, when multiplied together, yield a product of 1. In the context of solving the equation \(\frac{m}{-5} = -5\), the reciprocal of \(\frac{1}{-5}\) is \( -5\), because \(\frac{1}{-5} \times -5 = 1\).

To isolate \(m\), we multiply both sides of the equation by \( -5\), which effectively cancels the denominator on the left-hand side and leaves \(m\) by itself. It's an indispensable operation in algebra that allows us to solve for variables quickly and neatly.