Problem 8 - Elementary Algebra

Question

Solve \(\frac{2 k^{2} m n}{5 p q}=-6 n\) for \(m\)

Step-by-Step Solution

Answer

Question: Solve the given equation for m: \(\frac{2 k^{2} m n}{5 p q}=-6 n\). Answer: \(m = \frac{-15 p q}{k^{2}}\).

Step 1: Cancel out common factor "n" on both sides

As we can see, there is a common factor "n" present on both sides of the equation. We can cancel out this common factor by dividing both sides by "n": \(\frac{2 k^{2} m n}{5 p q n} = \frac{-6 n}{n}\) which simplifies to \(\frac{2 k^{2} m}{5 p q} = -6\).

Step 2: Multiply both sides by (5pq)

In order to get rid of the fraction, we can multiply both sides of the equation by the denominator "5pq". This will eliminate the fraction on the left side: \((5 p q) \times \frac{2 k^{2} m}{5 p q} = -6 \times (5 p q)\). Notice that the factors \(5pq\) on the left side cancel each other out and we are left with \(2 k^2 m = -30 p q\).

Step 3: Solve for "m"

The final step is to isolate "m" on one side of the equation. We do this by dividing both sides by \(2k^2\): \(m = \frac{-30 p q}{2 k^{2}}\).

Step 4: Simplify the expression

We can simplify this expression further by dividing the numerator and the denominator by the greatest common divisor (GCD), which is 2: \(m = \frac{-15 p q}{k^{2}}\). Now, we have successfully solved the equation for \(m\), and we have found that \(m = \frac{-15 p q}{k^{2}}\).

Key Concepts

Equation Solving

Equation solving in algebra involves making one variable the subject of the equation by manipulating both sides in a balanced way. When we encounter an equation like \( \frac{2 k^{2} m n}{5 p q}=-6 n \), our goal is to isolate the variable we need to solve for, in this case, \( m \).
To start, look for terms that appear on both sides of the equation and can be canceled out. Here, the term \( n \) is common, simplifying our task. Once canceled, the equation becomes \( \frac{2 k^{2} m}{5 p q} = -6 \).
Each step should maintain the equality of both sides, which is the essence of equation solving. Balancing means performing the same operation on both sides, like when we cancel \( n \) or multiply by \( 5pq \) to eliminate the fraction. By gradually isolating \( m \), we simplify the expression systematically until we reach the solution.

Fraction Simplification

Simplifying fractions makes equations much easier to work with. When dealing with \( \frac{2 k^{2} m n}{5 p q} = -6 n \), the first simplification step involved canceling the common factor \( n \) from both sides.
This results in \( \frac{2 k^{2} m}{5 p q} = -6 \), which is simpler than the original fraction.
The next step involves another form of simplification: clearing the fraction by multiplying both sides by the denominator \( 5pq \).

• This action turns the fraction on the left side into a simple expression: \( 2 k^2 m = -30 pq \).
• By simplifying, we prevent fractions from complicating the rest of the algebra operations, making it straightforward to solve for the variable.

Remember, a fraction is simplified when you can't make it smaller by canceling terms or factors. This rule will help keep your work neat and accurate.

Variable Isolation

Variable isolation is the final part of solving our equation. Here, the goal is to have \( m \) alone on one side of the equation. After we have \( 2 k^2 m = -30 pq \), isolating \( m \) involves dividing both sides by the coefficient \( 2k^2 \).
This results in \( m = \frac{-30 pq}{2 k^2} \).
Finally, we simplify the right side of the equation. Divide both the numerator and the denominator by their greatest common divisor, which is 2.

• This gives us \( m = \frac{-15 pq}{k^2} \), where now \( m \) is isolated and neatly expressed.
• The process of isolating a variable typically involves reversing the order of operations used to combine terms and simplify them, always aiming to maintain balance in the equation.

It's like unwinding a series of math operations to re-build the equation with the variable clearly defined.