Problem 56 - Algebra for College Students

Question

Solve each equation. $$\frac{2 m}{5}=\frac{10}{m}$$

Step-by-Step Solution

Answer

The solutions are m = 5 and m = -5.

Step 1: Set up the equation

The given equation is \[ \frac{2m}{5} = \frac{10}{m} \] To eliminate the fractions, multiply both sides of the equation by the common denominator, which is 5m.

Step 2: Multiply both sides by 5m

Multiplying both sides by 5m gives: \[ 5m \cdot \frac{2m}{5} = 5m \cdot \frac{10}{m} \] Simplify both sides.

Step 3: Simplify the equation

Simplify the left-hand side: \[ 5m \cdot \frac{2m}{5} = 2m^2 \] Simplify the right-hand side: \[ 5m \cdot \frac{10}{m} = 50 \] The equation now becomes: \[ 2m^2 = 50 \]

Step 4: Solve for m

To solve for m, divide both sides by 2: \[ m^2 = 25 \] Take the square root of both sides: \[ m = \pm 5 \]

Key Concepts

Common Denominator

When dealing with rational equations, finding a common denominator is a vital step. This makes it easier to combine and simplify fractions.
In our problem, we have the equation \(\frac{2m}{5} = \frac{10}{m}\).
The denominators here are 5 and m. To eliminate these denominators, find the least common multiple (LCM) of 5 and m, which is 5m.

• By multiplying both sides of the equation by 5m, you eliminate the fractions. This simplifies the equation and makes it easier to solve.
• This process involves distributing and then simplifying, which leads to a much more straightforward equation without fractions.

Simplifying Equations

Simplifying equations is crucial for making them easier to solve. In our example, we started by multiplying both sides of the equation by 5m to get:
\[ 5m \times \frac{2m}{5} = 5m \times \frac{10}{m} \]

• On the left-hand side, the 5s cancel out, leaving us with 2m\textsuperscript{2}.
• On the right-hand side, the m's cancel out, leaving us with 50.

Now, our equation is simplified to: \[ 2m\textsuperscript{2} = 50 \]
This is much easier to work with.
By simplifying, we create equations that are more straightforward and manageable. It involves removing any fractions or complex numbers, making the equation as simple as possible.

Square Root

Taking the square root is a frequent step in solving quadratic equations. After simplifying, our problem became: \[ 2m^2 = 50 \]
To solve for m, first divide both sides by 2, yielding: \[ m^2 = 25 \]

• Next, take the square root of both sides. The square root of both sides must be taken to find the value for 'm'.
• This gives us: \[ m = \pm 5 \] meaning 'm' can be either 5 or -5.

Square roots are essential in solving quadratics because they reverse the squaring process.
Remember: when taking the square root of both sides, always consider both the positive and negative roots.