Chapter 8.7, Problem 52HP

(a)

To determine

The equation $n=\frac{x+y-1}{xy}$ for x .

Answer to Problem 52HP

  $x=\frac{y-1}{ny-1}$

Explanation of Solution

Given information:

The equation: $n=\frac{x+y-1}{xy}$ .

Concept Used:

• To get rid of a number in addition from one side, subtract the same number from both sides of equal sign.
• To get rid of a number in subtraction from one side, add the same number both sides of equal sign.
• To get rid of a number in multiplication from one side, divide the same number from both sides of equal sign.
• To get rid of a number in division from one side, multiply the same number both sides of equal sign.

Rules of Addition/ Subtraction:

• Two numbers with similar sign always get added and the resulting number will carry the similar sign.
• Two numbers with opposite signs always get subtracted and the resulting number will carry the sign of larger number.

Rules of Multiplication/ Division:

• The product/quotient of two similar sign numbers is always positive.
• The product/quotient of two numbers with opposite signs is always negative.

Calculation:

In order to solve the formula for x , first multiply both sides by xy, then subtract x from both sides and then divide both sides by (ny-1) to simplify further as shown below:

  $\begin{array}{l}n=\frac{x+y-1}{xy}\\ n\times xy=\frac{x+y-1}{xy}\times xy\\ nxy=x+y-1\\ nxy-x=x-x+y-1\\ x\left(ny-1\right)=y-1\\ \frac{x\left(ny-1\right)}{\left(ny-1\right)}=\frac{y-1}{ny-1}\\ x=\frac{y-1}{ny-1}\end{array}$

Thus, the formula for the variable is $x=\frac{y-1}{ny-1}$ .

(b)

To determine

The equation $\frac{x+y}{x-y}=\frac{1}{2}$ for y .

Answer to Problem 52HP

  $y=\frac{-x}{3}$

Explanation of Solution

Given information:

The equation: $\frac{x+y}{x-y}=\frac{1}{2}$ .

Concept Used:

• To get rid of a number in addition from one side, subtract the same number from both sides of equal sign.
• To get rid of a number in subtraction from one side, add the same number both sides of equal sign.
• To get rid of a number in multiplication from one side, divide the same number from both sides of equal sign.
• To get rid of a number in division from one side, multiply the same number both sides of equal sign.

Rules of Addition/ Subtraction:

• Two numbers with similar sign always get added and the resulting number will carry the similar sign.
• Two numbers with opposite signs always get subtracted and the resulting number will carry the sign of larger number.

Rules of Multiplication/ Division:

• The product/quotient of two similar sign numbers is always positive.
• The product/quotient of two numbers with opposite signs is always negative.

CALCULATION:

In order to solve for the variable y , first multiply both sides by (x-y), then again multiply both sides by 2, then subtract 2x from both sides , then add y both sides and then divide both sides by 3 to simplify further as shown below:

  $\begin{array}{l}\frac{x+y}{x-y}=\frac{1}{2}\\ \frac{x+y}{x-y}\times \left(x-y\right)=\frac{1}{2}\times \left(x-y\right)\\ x+y=\frac{1}{2}\left(x-y\right)\\ 2x+2y=2\times \frac{1}{2}\left(x-y\right)\\ 2x+2y=x-y\\ 2x-2x+2y=x-2x-y\\ 2y=-x-y\\ 2y+y=-x-y+y\\ 3y=-x\\ \frac{3y}{3}=\frac{-x}{3}\\ y=\frac{-x}{3}\end{array}$

Thus, the formula for the variable is $y=\frac{-x}{3}$ .