Problem 77 - Elementary Algebra

Question

For the following problems, solve the rational equations. $$ \frac{6 a}{7}+\frac{2 a-3}{21}=\frac{77}{21} $$

Step-by-Step Solution

Answer

Short Answer: The solution to the rational equation is a = 4. To solve it, we first found the lowest common denominator (LCD) of 7 and 21, which is 21. We then cleared the fractions by multiplying each term by the LCD and simplified the resulting equation. Finally, we solved for the variable 'a' and verified the solution by substituting the value back into the original equation.

Step 1: (Step 1: Identify the lowest common denominator)

(Identify the lowest common denominator for the fractions in the equation. The denominators are 7 and 21, and the LCD is 21.)

Step 2: (Step 2: Clear the fractions)

(Multiply each term in the equation by the LCD (21). This will result in an equation without fractions. $$ 21\left(\frac{6 a}{7}\right)+21\left(\frac{2 a-3}{21}\right)=21\left(\frac{77}{21}\right) $$)

Step 3: (Step 3: Simplify the equation)

(Now, simplify the equation by canceling out factors and combining like terms. $$ (3)(6 a)+(2 a-3)=(1)(77) $$ $$ 18a+2a-3=77 $$)

Step 4: (Step 4: Solve for the variable 'a')

(Combine the a terms and isolate the variable on one side of the equation. $$ 20a-3=77 \\ 20a=80 \\ a=4 $$)

Step 5: (Step 5: Verify the solution)

(Substitute the value of 'a' back into the original equation to make sure it satisfies the equation. $$ \frac{6(4)}{7}+\frac{2(4)-3}{21}=\frac{77}{21} \\ \frac{24}{7}+\frac{5}{21}=\frac{77}{21} $$ Multiplying both sides by 21: $$ (3)(24)+5=77 \\ 72+5=77 \\ 77=77 $$ The solution is verified.)

Key Concepts

Lowest Common Denominator

When solving rational equations, identifying the Lowest Common Denominator (LCD) is crucial. It helps in transforming the equation by eliminating fractions, making it easier to solve. In the given exercise, the denominators are 7 and 21. The LCD for these numbers is 21. Finding the LCD involves determining the smallest number that both denominators can evenly divide into.

• Step 1: List the multiples of each denominator.
• Step 2: Identify the smallest common multiple.

In this example, multiples of 7 are 7, 14, 21, 28, 35, ... and so on, while multiples of 21 are 21, 42, 63, ..., etc. Therefore, 21 is the smallest number common to both, making it the LCD.

Simplifying Fractions

Simplifying fractions is about rewriting them in their simplest form, or reducing them. Once you have the LCD, you can use it to clear fractions by multiplying each term in the equation by the LCD. Our initial step in clearing fractions is essential for simplifying the equation:\[21\left(\frac{6 a}{7}\right)+21\left(\frac{2 a-3}{21}\right)=21\left(\frac{77}{21}\right)\]This step involves using the properties of fractions:

• The fraction \(\frac{6a}{7}\) becomes \((3)(6a)\) after multiplication, as 21 divided by 7 equals 3.
• The fraction \(\frac{2a-3}{21}\) simplifies as \(2a-3\), since 21 divided by 21 is 1.

This transformation ensures that an equation without fractions is achieved, which is easier to handle in subsequent steps.

Solving for Variables

In solving rational equations, isolating the variable is a pivotal step. It often involves combining like terms and then performing operations to isolate the variable on one side of the equation. After clearing fractions, the simplified equation is:\[18a + 2a - 3 = 77\]Here, the goal is to simplify by combining like terms:

• Combine the terms containing the variable to get \(20a\).
• Eliminate constant terms by adding or subtracting them from both sides, arriving at \(20a = 80\).
• Finally, solve for \(a\) by dividing both sides by 20, resulting in \(a = 4\).

These systematic steps are key to effectively solving the equation for the unknown variable.

Verifying Solutions

Verification is an important final step in checking solutions. After finding the variable's value, substitute it back into the original equation to ensure it holds true. This reassures that no computational errors were made. For \(a = 4\), substitute back:\[\frac{6(4)}{7} + \frac{2(4)-3}{21} = \frac{77}{21}\]Simplify each term:

• The first term becomes \(\frac{24}{7}\).
• The second term simplifies to \(\frac{5}{21}\).

Check if these add up to the right-hand side of the original equation:

• Multiply by the LCD to see if both sides equal. They do: \(77 = 77\).

This calculation confirms the solution's validity, reinforcing confidence in the solved value.