Chapter 10, Problem 23SGR

To determine

The equation $\frac{1}{x^2}+\frac{1}{y}=4$ for y .

Answer to Problem 23SGR

  $y=4-x^2$

Explanation of Solution

Given information:

The equation is: $\frac{1}{x^2}+\frac{1}{y}=4$ .

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.

Calculationy:

In order to solve the formula for y , the given equation can be written as $\frac{y+x^2}{x^{2}y}=4$ . first multiply both sides by x²y, then subtract both sides by x², then again subtract 4x²y from both sides and then divide both sides by (1- x ² y) to simplify further as shown below:

  $\begin{array}{l}\frac{1}{x^2}+\frac{1}{y}=4\\ \frac{y+x^2}{x^{2}y}=4\\ \frac{x^{2}y\left(y+x^2\right)}{x^{2}y}=4x^{2}y\\ y+x^2=4x^{2}y\\ y+x^2-x^2=4x^{2}y-x^2\\ y=4x^{2}y-x^2\\ y-4x^{2}y=4x^{2}y-4x^{2}y-x^2\\ y\left(1-4x^2\right)=-x^2\\ \frac{y\left(1-4x^2\right)}{\left(1-4x^2\right)}=\frac{-x^2}{1-4x^2}\\ y=\frac{-x^2}{1-4x^2}\end{array}$

Thus, the formula is $y=\frac{-x^2}{1-4x^2}$ .