Problem 39 - Elementary Algebra

Question

In the following problems, solve each of the conditional equations. $$ \frac{3 m}{10}=-1 $$

Step-by-Step Solution

Answer

Answer: $$m = -\frac{10}{3}$$

Step 1: Observe the Given Equation

We have the equation: $$\frac{3m}{10} = -1$$. Our main goal is to solve for m.

Step 2: Eliminate the Fraction

In order to isolate m, we need to eliminate the fraction. We can accomplish this by multiplying both sides of the equation by the denominator, 10: $$(10)\cdot\Big(\frac{3m}{10}\Big) = (-1)\cdot(10)$$

Step 3: Simplify

Now, let's simplify both sides of the equation. On the left side, the 10's will cancel each other out, leaving us with: $$3m = -10$$

Step 4: Isolate the Variable m

Finally, to solve for m, we need to divide both sides of the equation by the coefficient of m, which is 3: $$\frac{3m}{3} = \frac{-10}{3}$$

Step 5: Find the Value of m

Simplifying the equation, we get the solution for m: $$m = -\frac{10}{3}$$

Key Concepts

Understanding Fractions in Equations

Fractions can be intimidating at first, but they are simply a way to represent division.When solving equations like \(\frac{3m}{10} = -1\), it's important to understand that the fraction \(\frac{3m}{10}\) means \(3m\) is divided by 10.To eliminate the fraction, you use the inverse operation of division, which is multiplication.By multiplying both sides of the equation by 10, you effectively remove the fraction:

• This means, \(10 \times \frac{3m}{10} = (-1)\times10\)
• After simplification, this operation cancels the 10 below, leaving you with \(3m = -10\)

Eliminating fractions in equations allows for simpler solutions and a clearer path to isolating variables.

The Role of Coefficients

In algebra, a coefficient is the number placed before a variable.In our equation, \(3\) is the coefficient of \(m\).Coefficients tell us how many times the variable is multiplied. When you need to solve for the variable, knowing the coefficient is crucial because

• It guides us on how to isolate the variable;
  • in this case, by dividing both sides of the equation by the coefficient.

So, after removing the fraction, we divide \(3m = -10\) by 3:

• This will give \(m\) alone: \(m = \frac{-10}{3}\)

Recognizing and using coefficients correctly is key to efficiently solving algebraic equations.

Isolating the Variable

Isolating the variable is a main step in solving any algebraic equation.The process involves rearranging the equation so that the unknown variable, in this case, \(m\), is on one side of the equation by itself.In order to isolate \(m\) in the equation \(3m = -10\), we use the inverse operation to undo the multiplication of 3.

• That means dividing each side by 3, leading us to \(m = \frac{-10}{3}\)

Isolating the variable allows us to express the solution in the simplest and most understandable form possible.

Algebraic Equations and Their Solutions

Algebraic equations involve using mathematical symbols and operations to represent real-world problems or abstract scenarios.Solving an equation means finding the value of the variable that makes the equation true.In our given equation, \(\frac{3m}{10} = -1\), we underwent a series of steps to isolate \(m\).To solve an algebraic equation successfully:

• Eliminate fractions by multiplying each term by the denominator.
• Proceed by removing coefficients next to the variable using inverse operations.
• Continue these steps until the variable is isolated and a straightforward solution is achieved.

Understanding and following these step-by-step strategies helps to solve any algebraic equation systematically and ensures accuracy in finding solutions.