Question

Solve each equation, if possible. $$ \frac{2}{y}+\frac{4}{y}=3 $$

Short answer

y = 2

Step-by-step solution

Combine Like Terms

Combine the fractions on the left side of the equation. The fractions have the same denominator, so simply add the numerators together: \[ \frac{2 + 4}{y} = 3 \rightarrow \frac{6}{y} = 3 \]

Isolate y

To isolate the variable, multiply both sides of the equation by y to get rid of the fraction: \[ 6 = 3y \]

Solve for y

Divide both sides of the equation by 3 to solve for y: \[ y = \frac{6}{3} \rightarrow y = 2 \]

Verify the Solution

Substitute the solution back into the original equation to verify: \[ \frac{2}{2} + \frac{4}{2} = 1 + 2 = 3 \] Since both sides of the equation are equal, the solution is verified.

Key concepts

Combining Like Terms

In this step-by-step solution, we start by combining like terms. This is a fundamental concept in algebra that simplifies an equation by adding or subtracting terms with the same variable. For example, in the equation \(\frac{2}{y} + \frac{4}{y} = 3\), we have two rational terms with the same denominator \(y\).

By adding the numerators, we simplify the left side of the equation to \(\frac{6}{y} = 3\). This combining process makes the equation simpler and easier to solve. Always look for terms with the same variable and combine them to reduce complexity.

Isolating Variables

After combining like terms, the next step is isolating the variable. This means we want to get the variable (in this case, \(y\)) by itself on one side of the equation.

In the equation \(\frac{6}{y} = 3\), multiplying both sides of the equation by \(y\) removes the fraction, giving us \(6 = 3y\). This step is crucial as it transforms a rational equation into a simpler format that is easier to handle.

Make sure to perform the same operation (like multiplication or division) on both sides of the equation to maintain equality.

Verifying Solutions

Once we have solved for the variable, it's important to verify the solution by substituting it back into the original equation. In our exercise, we found \(y = 2\).

We substitute this value back into the initial equation: \(\frac{2}{2} + \frac{4}{2} = 3\). Calculating the left side, we get \(1 + 2 = 3\), which matches the right side.

Verifying solutions helps us confirm the correctness of our answer. Always perform this step to ensure that no mistakes were made during the solving process.

Basic Algebraic Operations

Solving rational equations often involves fundamental algebraic operations like addition, multiplication, and division.

In our example, we added fractions, multiplied both sides by \(y\), and divided by 3.

Understanding these basic operations is essential for simplifying and solving equations. Always remember to do the same operation on both sides of the equation to keep it balanced.

Mastering these core operations builds a strong foundation for tackling more complex algebraic problems.