Problem 10 - Algebra 2

Question

Solve each equation or inequality. Check your solutions. $$ \frac{y}{y+1}=\frac{2}{3} $$

Step-by-Step Solution

Answer

The solution is \( y = 2 \).

Step 1: Set Up the Equation

The original equation given is \( \frac{y}{y+1} = \frac{2}{3} \). We need to solve this equation for \( y \).

Step 2: Cross-Multiply

To eliminate the fractions, we cross-multiply. This means we multiply the numerator of one fraction by the denominator of the other and set the expressions equal to each other: \( 3y = 2(y+1) \).

Step 3: Distribute and Simplify

Distribute the 2 on the right side: \( 3y = 2y + 2 \).

Step 4: Isolate the Variable

Subtract \( 2y \) from both sides to get \( y = 2 \).

Step 5: Check the Solution

Substitute \( y = 2 \) back into the original equation to verify: \( \frac{2}{2+1} = \frac{2}{3} \rightarrow \frac{2}{3} = \frac{2}{3} \). The solution is correct.

Key Concepts

Cross-Multiplication

To solve equations involving fractions, one effective method is cross-multiplication. This technique eliminates the fractions.When you have an equation in the form \( \frac{a}{b} = \frac{c}{d} \), you perform cross-multiplication by multiplying the numerator of the first fraction by the denominator of the second fraction and vice versa.So, for our example equation \( \frac{y}{y+1} = \frac{2}{3} \), we need to multiply \( y \) by 3 and \( 2 \) by \( y+1 \).

• This gives us the equation \( 3y = 2(y+1) \).
• Cross-multiplication helps clear fractions, making the equation easier to solve.

Variable Isolation

Once you have an equation free of fractions, the next goal is to isolate the variable. This means getting the variable \( y \) by itself on one side of the equation.In our example, after cross-multiplication, we have the equation \( 3y = 2(y+1) \).

• First, distribute the 2 on the right side: \( 2 \times (y+1) = 2y + 2 \).
• This simplifies the equation to \( 3y = 2y + 2 \).
• Next, subtract \( 2y \) from both sides to remove it from the right side, resulting in \( y = 2 \).

By arranging the equation so \( y \) stands alone, you've successfully isolated the variable. This step is crucial because it reveals the value of the unknown part you're solving for.

Equation Verification

After finding a solution for the variable, it's essential to verify that the solution is correct. This process ensures that your answer satisfies the original equation.For the equation \( \frac{y}{y+1} = \frac{2}{3} \), we determined that \( y = 2 \).

• Substitute \( y = 2 \) back into the original equation.
• This gives us \( \frac{2}{2+1} = \frac{2}{3} \).
• Simplifying the left side results in \( \frac{2}{3} \).

Since both sides equal \( \frac{2}{3} \), the solution \( y = 2 \) is verified as correct.Feedback loops like checking solutions are a critical step in problem-solving, ensuring both accuracy and understanding.