Question $3+\frac{5}{2 p}=\frac{4}{p}$

Studdy Solution

STEP 1

What is this asking? We need to find the value of $p$ that makes this equation true! Watch out! Remember, we can't divide by zero, so keep an eye on that $p$ in the denominators!

STEP 2

1. Prepare the equation
2. Isolate the variable
3. Find the solution

STEP 3

Alright, let's get rid of those fractions!
We want to work with whole numbers whenever possible.
The least common denominator of 2 $p$ and $p$ is 2 $p$ .
So, we're going to multiply both sides of the equation by 2 $p$ .
Remember, what we do to one side, we must do to the other to keep the equation balanced!

STEP 4

$2p \cdot \left( 3 + \frac{5}{2p} \right) = 2p \cdot \left( \frac{4}{p} \right)$

STEP 5

Now, distribute that 2 $p$ on the left side: $2p \cdot 3 + 2p \cdot \frac{5}{2p} = 2p \cdot \frac{4}{p}$

STEP 6

Let's simplify!
On the left, $2p \cdot 3$ becomes 6 $p$ .
Then, $2p \cdot \frac{5}{2p}$ becomes 5 (since the 2 $p$ in the numerator and the 2 $p$ in the denominator divide to one).
On the right side, $2p \cdot \frac{4}{p}$ becomes 8 (since $p$ divides to one). $6p + 5 = 8$

STEP 7

Let's isolate $p$ !
We want to get $p$ all by itself on one side of the equation.
We have a positive 5 added to 6 $p$ , so to move it to the other side, we'll add negative 5 to both sides of the equation. $6p + 5 + (-5) = 8 + (-5)$

STEP 8

This simplifies to: $6p = 3$

STEP 9

Now, $p$ is being multiplied by 6.
To undo this and get $p$ by itself, we'll divide both sides by 6. $\frac{6p}{6} = \frac{3}{6}$

STEP 10

Simplify! On the left, the 6's divide to one, leaving just $p$ .
On the right, 3 divided by 6 simplifies to $\frac{1}{2}$ . $p = \frac{1}{2}$

STEP 11

So, the value of $p$ that makes the original equation true is $p = \frac{1}{2}$ !

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