Problem 65 - Elementary Algebra Within Reach

Question

In Exercises 65-68, solve the rational equation. $$ \frac{x}{5}+\frac{2 x}{5}=3 $$

Step-by-Step Solution

Answer

The solution to the rational equation is \(x = 5\).

Step 1: Combine Similar Fractions

The first step is to combine like terms on the left side of the equation, producing \(\frac{3x}{5} = 3\).

Step 2: Multiply by Common Denominator

Next step is to get rid of the fractions, which can be achieved by multiplying both sides of the equation by the common denominator, 5.\nDoing so, results in the equation \(3x = 15\).

Step 3: Solve for the Variable

To solve for \(x\), we must isolate it to one side of the equation. We achieve this by dividing both sides of the equation by 3, resulting in \(x = 5\).

Key Concepts

Combining Like Terms

When solving rational equations, the first important step often involves combining like terms. This means finding terms that have the same variable and simplifying them as a single term.
In the given rational equation \(\frac{x}{5} + \frac{2x}{5} = 3\), we notice that both terms on the left contain the variable \(x\) with the same denominator 5.

• Since the denominators are identical, these terms are "like" terms.
• To combine them, simply add their numerators together.

This gives us \(\frac{1x + 2x}{5} = \frac{3x}{5}\), effectively reducing the equation to a simpler form, \(\frac{3x}{5} = 3\).
By combining like terms, we make the equation easier to manipulate in subsequent steps.

Common Denominator

Rational equations often involve fractions, and dealing with them effectively requires finding a common denominator.
In our case, after combining like terms, we are left with \(\frac{3x}{5} = 3\).

• The common denominator, in this case, is 5, which we use to eliminate the fraction.
• This process transforms the equation into a simpler linear equation, making it easier to solve for the variable.

Multiply every term in the equation by 5, the least common multiple of the denominators, to eliminate the fractions:\[ 3x = 15 \] Removing the fractions makes the equation straightforward, setting the stage for the final step of solving for the variable.

Solving for Variables

Once you have simplified the equation and eliminated fractions, the next step is solving for the variable.
We simplified our rational equation to \(3x = 15\).

• The goal now is to isolate the variable \(x\) on one side of the equation.
• To achieve this, divide both sides by the coefficient of \(x\) (which is 3).

By performing this division:\[ \frac{3x}{3} = \frac{15}{3} \] You solve for \(x\), finding that \(x = 5\).
The equation is now solved, and through isolating the variable, we have successfully found the solution.