# What is the value of p in 2/x + p = 3 if (5(7x+5))/3 - 23/2 = 13?

Jun 9, 2018

$p = \frac{151}{97} \approx 1.56$

#### Explanation:

First solve for the value of $x$

$\frac{5 \left(7 x + 5\right)}{3} - \frac{23}{2} = 13$

$\frac{35 x + 25}{3} - \frac{23}{2} = 13$

use a GCD to remove the fractions:

$6 \left[\frac{35 x + 25}{3} - \frac{23}{2} = 13\right]$

$2 \left(35 x + 25\right) - 3 \left(23\right) = 6 \left(13\right)$

$70 x + 50 - 69 = 78$

$70 x - 19 = 78$

$70 x = 97$

$x = \frac{97}{70}$

Now we solve for $p$:

$\frac{2}{x} + p = 3$

$\frac{2}{\frac{97}{70}} + p = 3$

$\frac{140}{97} + p = 3$

$p = 3 - \frac{140}{97}$

$p = \frac{151}{97} \approx 1.56$

Jun 9, 2018

Using first principles
$p = \frac{151}{97}$

$x = \frac{97}{70}$

#### Explanation:

In school it is good practice to explain what steps your are applying. That way the teacher can see your way of thinking about manipulation and better understand you intention.

Given:
$\frac{2}{x} + p = 3 \text{ } \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . E q u a t i o n \left(1\right)$

$\frac{5 \left(7 x + 5\right)}{3} - \frac{23}{2} = 13 \text{ } \ldots \ldots \ldots \ldots . . E q u a t i o n \left(2\right)$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$E q u a t i o n \left(1\right)$ has 2 unknowns so can not be solved directly. We need 1 equation with 1 unknown. That situation exists in $E q u a t i o n \left(2\right)$

So we can solve for $x$ in $E q n \left(2\right)$ and then substitute for $x$ in $E q n \left(1\right)$. Thus solving for $p$.

$\textcolor{b r o w n}{\text{Consider } E q u a t i o n \left(2\right) \to \frac{5 \left(7 x + 5\right)}{3} - \frac{23}{2} = 13}$

Add $\frac{23}{2}$ to both sides giving:

$\frac{5 \left(7 x + 5\right)}{3} = \frac{49}{2}$

Multiply both sides by $\frac{3}{5}$

$7 x + 5 = \frac{3}{5} \times \frac{49}{2}$

$7 x + 5 = \frac{147}{10}$

Subtract 5 from both sides:

$7 x = \frac{97}{10}$

Divide both sides by 7

$\textcolor{b l u e}{x = \frac{97}{70}}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b r o w n}{\text{Substitute for "x" in } E q u a t i o n \left(1\right)}$

$\textcolor{g r e e n}{\frac{2}{\textcolor{red}{x}} + p = 3 \textcolor{w h i t e}{\text{dddd") ->color(white)("dddd}} \left(2 \div \textcolor{red}{\frac{97}{70}}\right) + p = 3}$

$\textcolor{g r e e n}{\textcolor{w h i t e}{\text{ddddddddddd.d")->color(white)("ddddddd")140/97color(white)("dd}} + p = 3}$

Subtract $\frac{140}{97}$ from both sides

$\textcolor{b l u e}{\textcolor{w h i t e}{\text{ddddddddddddd")->color(white)("dddddd}} p = \frac{151}{97}}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b r o w n}{\text{Check}}$

$p = \frac{151}{97}$

$x = \frac{97}{70}$

Left hand side of $\frac{2}{x} + p = 3$

$\frac{\textcolor{w h i t e}{. .} 2 \textcolor{w h i t e}{. .}}{\frac{97}{70}} + \frac{151}{97}$

$\frac{140}{97} + \frac{151}{97}$

$\frac{291}{97} \to 3$

Thus $L H S = R H S = 3$