Problem 14 - Algebra for College Students

Question

$$\text { Solve each formula for the indicated variable.}$$ $$\frac{1}{x}-\frac{2}{3 y}+z=0 \text { for } y$$

Step-by-Step Solution

Answer

y = \frac{2}{3 (\frac{1}{x} + z)}

Step 1: Isolate the term with y

Start by isolating \( -\frac{2}{3y} \) on one side of the equation. Rewrite the given formula: \( \frac{1}{x} - \frac{2}{3y} + z = 0 \). Move \( \frac{1}{x} \) and \( z \) to the other side: \( -\frac{2}{3y} = -\frac{1}{x} - z \).

Step 2: Eliminate the negative sign

Multiply both sides of the equation by \( -1 \) to simplify: \( \frac{2}{3y} = \frac{1}{x} + z \).

Step 3: Solve for y

To solve for \( y \), first rearrange the equation: \( 2 = (\frac{1}{x} + z) \times 3y \). Divide both sides by \( 3 (\frac{1}{x} + z) \): \( y = \frac{2}{3 (\frac{1}{x} + z)} \).

Key Concepts

algebraic manipulation

Algebraic manipulation involves rearranging and simplifying equations to make them easier to solve. It is a fundamental technique in algebra, used to isolate variables or simplify complex expressions.
One common method of algebraic manipulation is to move terms from one side of the equation to the other. This technique is used to gather like terms and simplify the equation.
Another technique is to multiply or divide both sides of the equation by the same number to eliminate fractions or coefficients. This simplifies the equation further, making it easier to isolate the desired variable.
Understanding these basic skills is crucial because they allow you to solve more complex problems. Learning how to manipulate equations correctly helps to build a strong foundation in algebra.
If we look at our exercise, the first step involved moving terms around to isolate the term with y. We moved \( \frac{1}{x} \) and z to the other side of the equation, which is a common algebraic manipulation.

solving equations

Solving equations means finding the value of the variable that makes the equation true. In algebra, equations can be solved by using a variety of methods including simple operations and more complex strategies.
There are basic steps to follow when solving an equation:

• Simplify both sides of the equation, if necessary.
• Isolate the variable term on one side of the equation.
• Perform the same operations on both sides of the equation to maintain equality.
• Solve for the variable by performing final operations like division or multiplication.

For example, in our exercise, we started with \( \frac{1}{x} - \frac{2}{3y} + z = 0 \). By isolating \( -\frac{2}{3y} \) and then eliminating the negative sign by multiplying both sides by -1, we simplified our equation which made it easier to solve.

isolating variables

Isolating variables is the process of getting the variable you need to solve for by itself on one side of the equation.
This usually involves several steps of algebraic manipulation to move all other terms to the opposite side of the equation.
Here are some key tips for isolating variables:

• Identify the term that contains the variable you want to isolate.
• Use addition or subtraction to move other terms away from the variable term.
• Eliminate any coefficients or fractions around the variable by using multiplication or division.
• Check your work by substituting your solution back into the original equation.

In our problem, once we isolated \( -\frac{2}{3y} \), we had to manipulate the terms around y. We multiplied both sides by -1 and then rearranged the equation to solve for y.Finally, we divided both sides by 3( \( \frac{1}{x} + z \) which isolated y resulting in our final expression:
\( y = \frac{2}{3 (\frac{1}{x} + z)} \).