Problem 51 - Algebra 2

Question

Solve each equation or formula for the specified variable. $$ x=\frac{y}{y+4}, \text { for } y $$

Step-by-Step Solution

Answer

The solution for y is \( y = \frac{4x}{1-x} \).

Step 1: Clear the Fraction

Start by multiplying both sides of the equation by the denominator to get rid of the fraction: \[ x(y+4) = y \]

Step 2: Distribute the x

Distribute the x on the left side: \[ xy + 4x = y \]

Step 3: Rearrange Terms

Subtract xy from both sides to start isolating y: \[ 4x = y - xy \]

Step 4: Factor out y

Factor y out of the right-hand side:\[ 4x = y(1 - x) \]

Step 5: Solve for y

Divide both sides by the factor (1-x) to solve for y:\[ y = \frac{4x}{1-x} \]

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the value of a variable that makes an equation true. This process relies on manipulating the equation using different algebraic techniques. To solve an equation, follow these basic steps:

• Identify the operations performed on the variable.
• Use inverse operations to undo these, aiming to isolate the variable.
• Maintain balance by performing the same operation on both sides of the equation.

Maintaining balance in the equation is crucial. If you alter one side, the equation will only hold if you apply the same change to the opposite side. This way, the relationship between the quantities remains the same, leading you to the correct solution.

Fractions

Fractions often appear in equations and can complicate solving them. They represent a part of a whole and consist of a numerator (top number) and a denominator (bottom number). When solving equations with fractions, one common technique is to eliminate them as a first step.You can clear fractions by:

• Multiplying every term in the equation by the least common denominator (LCD).
• This removes the fractions, simplifying the process of solving the equation.

In the given exercise, the fraction \( \frac{y}{y+4} \) caused complexity. We cleared it by multiplying both sides by \( y+4 \), thus transforming the equation into a simpler form without fractions.

Variable Isolation

Variable isolation is the process of rearranging an equation to express a variable independently on one side. This is often the ultimate goal when solving equations as it gives us the solution of the problem.To isolate a variable:

• Perform operations to remove any numbers or other variables on the same side as the target variable.
• Make sure to do these operations on both sides to keep the equation balanced.

In the exercise, to solve for \( y \), we moved terms around to have all expressions involving \( y \) on one side. Continuing to manipulate these allowed us to express \( y \) independently, showing its value related to \( x \).

Distributive Property

The distributive property is a useful property of multiplication over addition. It allows us to distribute a multiplied factor across terms added or subtracted inside parentheses. The formula can be written as: \[ a(b + c) = ab + ac \]In the exercise, this property was employed when multiplying \( x \) by the terms inside the parentheses \( (y+4) \). This step ensured that the equation was suitably expanded and prepared for further isolation of \( y \).Using the distributive property:

• Makes equations easier to work with by expanding terms.
• Is often a preliminary step before isolating variables.

Mastering this property can streamline complex equations significantly, making solving them more strategic and straightforward.