Problem 76 - Elementary Algebra

Question

For the following problems, solve the rational equations. $$ \frac{4 x}{5}+\frac{3 x-1}{15}=\frac{29}{25} $$

Step-by-Step Solution

Answer

Answer: The solution to the given rational equation is \(x=\frac{92}{75}\).

Step 1: Find the Least Common Denominator (LCD)

First, we need to find the least common denominator (LCD) of the fractions in the given equation. In this case, the denominators are 5, 15, and 25. The least common denominator for these numbers is 75.

Step 2: Multiply each term by the LCD to clear fractions

Now, we will multiply each term by 75 to clear out the fractions in the equation. This will give us: $$ 75 \cdot \frac{4x}{5} + 75 \cdot \frac{3x-1}{15} = 75 \cdot \frac{29}{25} $$

Step 3: Simplify and cancel out

Simplify each term by canceling out any factors that are common to both the numerator and denominator. This gives us: $$ 15(4x) + 5(3x-1) = 3(29) $$

Step 4: Distribute and combine like terms

Distribute the values inside the brackets and combine like terms. We get: $$ 60x + 15x - 5 = 87 $$

Step 5: Combine and solve for x

Combine the x terms and add 5 to both sides of the equation to solve for x: $$ 75x = 92 $$ Then, divide both sides by 75: $$ x= \frac{92}{75} $$ So, the solution to the given rational equation is \(x=\frac{92}{75}\).

Key Concepts

Least Common Denominator

When working with rational equations, finding the Least Common Denominator (LCD) is essential for simplifying the problem. The LCD refers to the smallest common multiple shared by the denominators in the equation.

• In this problem, the denominators are 5, 15, and 25.
• To determine the LCD, we need to identify the smallest number that is divisible by each of these denominators.
• The LCD for 5, 15, and 25 is 75.

Finding the LCD allows us to clear fractions by converting them into whole numbers. By multiplying each term in the equation by the LCD, we effectively eliminate the denominators, making it simpler to solve.

Solving Equations

Once the equation is cleared of fractions, the next step is to solve the resulting equation. In this context:

• We started with the modified equation: \[15(4x) + 5(3x-1) = 3(29)\]
• Each term or expression within the equation represents a part of the overall solution.
• By solving for the variable, we determine the value of \(x\) that satisfies the equation.

Solving involves simplifying through distribution and then combining like terms to isolate the variable. We continue simplification until we arrive at the final value of the variable, which, in this case, is \(x = \frac{92}{75}\).

Distributive Property

The Distributive Property is a key algebraic principle used in this equation. It states that a number outside the parenthesis can be multiplied by each term within the parenthesis. This property is fundamental when sorting through expressions within an equation.

• In our solution, we used it to multiply 15 by \(4x\) and 5 by \((3x - 1)\).
• This results in: \(60x + 15x - 5\).

Understanding and applying the distributive property helps in breaking down complex expressions and in simplifying equations further.

Combining Like Terms

After using the distributive property, the next logical step involves combining like terms. This technique simplifies equations by adding or subtracting terms that have the same variables.

• In our simplified equation, \(60x + 15x - 5 = 87\), we notice that both \(60x\) and \(15x\) are like terms.
• Combining them results in \(75x\), simplifying our equation to \(75x - 5 = 87\).

Finally, solving for \(x\) becomes straightforward as we isolate the variable and solve the equation. Identifying and combining like terms is vital because it further reduces the problem, leading to quicker solutions.