We are given several equations and asked to isolate a specified variable in each. c) $\frac{ax}{y} = \frac{bc}{e}$ solve for $e$ d) $\frac{5x}{y} = a+e$ solve for $y$ e) $x+5 = \frac{3m}{y}$ solve for $y$ f) $\frac{2n}{x-4} = m$ solve for $x$ g) $\frac{x}{5-y} = m-1$ solve for $y$ h) $k-y = \frac{e-b}{x+a}$ solve for $x$

1. Problem Description

We are given several equations and asked to isolate a specified variable in each.

c) $\frac{ax}{y} = \frac{bc}{e}$ solve for $e$

d) $\frac{5x}{y} = a+e$ solve for $y$

e) $x+5 = \frac{3m}{y}$ solve for $y$

f) $\frac{2n}{x-4} = m$ solve for $x$

g) $\frac{x}{5-y} = m-1$ solve for $y$

h) $k-y = \frac{e-b}{x+a}$ solve for $x$

2. Solution Steps

c) $\frac{ax}{y} = \frac{bc}{e}$

Multiply both sides by $e$:

$\frac{axe}{y} = bc$

Multiply both sides by $y$:

$axe = bcy$

Divide both sides by $ax$:

$e = \frac{bcy}{ax}$

d) $\frac{5x}{y} = a+e$

Multiply both sides by $y$:

$5x = (a+e)y$

Divide both sides by $(a+e)$:

$y = \frac{5x}{a+e}$

e) $x+5 = \frac{3m}{y}$

Multiply both sides by $y$:

$y(x+5) = 3m$

Divide both sides by $(x+5)$:

$y = \frac{3m}{x+5}$

f) $\frac{2n}{x-4} = m$

Multiply both sides by $(x-4)$:

$2n = m(x-4)$

$2n = mx - 4m$

Add $4m$ to both sides:

$2n + 4m = mx$

Divide both sides by $m$:

$x = \frac{2n+4m}{m}$

$x = \frac{2n}{m} + 4$

g) $\frac{x}{5-y} = m-1$

Multiply both sides by $(5-y)$:

$x = (m-1)(5-y)$

$x = 5(m-1) - (m-1)y$

$(m-1)y = 5(m-1) - x$

Divide both sides by $(m-1)$:

$y = \frac{5(m-1) - x}{m-1}$

$y = \frac{5(m-1)}{m-1} - \frac{x}{m-1}$

$y = 5 - \frac{x}{m-1}$

h) $k-y = \frac{e-b}{x+a}$

Multiply both sides by $(x+a)$:

$(k-y)(x+a) = e-b$

Divide both sides by $(k-y)$:

$x+a = \frac{e-b}{k-y}$

Subtract $a$ from both sides:

$x = \frac{e-b}{k-y} - a$

3. Final Answer

c) $e = \frac{bcy}{ax}$

d) $y = \frac{5x}{a+e}$

e) $y = \frac{3m}{x+5}$

f) $x = \frac{2n+4m}{m}$ or $x = \frac{2n}{m}+4$

g) $y = 5 - \frac{x}{m-1}$

h) $x = \frac{e-b}{k-y} - a$