91Ó°ÊÓ

Fractions can make solving linear equations more complex, so it's often useful to eliminate them right away. In our exercise, the equation has fractions: \( \frac{3}{5} p + 2 = \frac{4}{5} p - 1 \). To eliminate the fractions, we look for the least common multiple (LCM) of the denominators. The denominators here are 5, so the LCM is also 5.

To clear out the fractions, we multiply every term in the equation by this LCM (5). Multiplying, we get:
\[ 5 \times \frac{3}{5} p + 5 \times 2 = 5 \times \frac{4}{5} p - 5 \times 1 \]
This simplifies to: \[3p + 10 = 4p - 5 \]
Now, we have an equation without fractions, making it easier to solve.

Isolating the variable is a key step in solving linear equations. It means getting the variable by itself on one side of the equation. In our simplified equation from the previous step: \[3p + 10 = 4p - 5 \]
We need to get all terms involving \( p \) on one side and constant terms on the other.

First, we subtract \(3p\) from both sides of the equation:
\[10 = 4p - 3p - 5 \]
This simplifies to:
\[10 = p - 5 \]
Next, we move the constant term (\(-5\)) by adding 5 to both sides:
\[10 + 5 = p \]
This simplifies to: \[p = 15\]
Now, the variable \( p \) is isolated, and we've found its value.

The least common multiple (LCM) helps simplify equations involving fractions. The LCM of a set of numbers is the smallest multiple they all share. In our exercise, determining the LCM of the denominators 5 helps in eliminating the fractions.

Here's how the LCM process works:

• List the multiples of each number. For 5, these are 5, 10, 15, 20, etc.
• Identify the smallest multiple common to both numbers. For 5, it is 5 itself.

Once identified, we multiply every term in the equation by the LCM. This clears the fractions, simplifying the equation.
In the given problem: \[ \frac{3}{5} p + 2 = \frac{4}{5} p - 1 \]
We use 5 as the LCM and multiply: \[5 \times \frac{3}{5} p + 5 \times 2 = 5 \times \frac{4}{5} p - 5 \times 1 \] This simplifies to: \[3p + 10 = 4p - 5\] Thus, the LCM helps eliminate fractions, facilitating a smoother solving process.