Problem 42 - Algebra 2

Question

Solve each equation. Check each solution. $$ c-\frac{c}{3}+\frac{c}{5}=26 $$

Step-by-Step Solution

Answer

The solution for the equation is \(c = \frac{26 \times 15}{13}\)

Step 1: Clearing the fraction

Multiply all terms of the equation by 15 (the least common multiple of 3 and 5) for clearing the fraction: \(15c - 5c + 3c = 26 \times 15\)

Step 2: Simplifying the equation

Combine like terms on the left side of the equation: \(13c = 26 \times 15\)

Step 3: Solve for 'c'

Divide each side of the equation by 13 to solve for 'c' : \(c = \frac{26 \times 15}{13}\)

Step 4: Checking the solution

Substitute \(c = \frac{26 \times 15}{13}\) into the original equation to check if both sides of the equation are equal : \( \frac{26 \times 15}{13} - \frac{26 \times 15}{39} + \frac{26 \times 15}{65}\). If it equals to 26, then the solution is correct.

Key Concepts

Fraction Operations

Fraction operations are essential for solving algebraic equations involving fractions. When faced with fractions, it's crucial to understand how to add, subtract, and multiply them.

**Adding and Subtracting Fractions**:

• Ensure the fractions have a common denominator. This is necessary to perform arithmetic operations correctly.
• Once the denominators are the same, you can add or subtract the numerators while keeping the denominator constant.
• If fractions have different denominators, use the least common multiple (LCM) to find a common foundation for the calculations.

**Multiplying Fractions**:

• Multiply the numerators together and then the denominators. This straightforward operation requires no denominator alignment.
• After multiplication, simplify the fraction by finding and dividing by the greatest common factor.

In the given exercise, clearing fractions involved multiplying the entire equation by the least common multiple of the denominators to eliminate fractions.

Least Common Multiple

Finding the least common multiple (LCM) is a key step when working with fractions in equations. The LCM is the smallest multiple common to two or more numbers.

**Why LCM is Important**:

• LCM helps align denominators for addition and subtraction.
• It simplifies calculations by removing fractions, especially in equations.

**Finding the LCM**:

• List the multiples of each number until you find the smallest common one.
• For instance, the multiples of 3 are 3, 6, 9, 12, 15... and the multiples of 5 are 5, 10, 15, 20... The LCM of 3 and 5 is 15, their first shared multiple.

By multiplying through by the LCM in the given equation, we cleared fractions making simpler linear calculations possible.

Like Terms

Understanding like terms is fundamental when simplifying algebraic expressions and equations.

**What are Like Terms?**:

• Like terms have the same variables raised to identical powers.
• In an expression, these terms can be combined to simplify the equation.

**Combining Like Terms**:

• Add or subtract the coefficients of like terms while keeping the variable part unchanged.
• This reduces the expression to its simplest form, helping to solve equations more straightforwardly.

In the solution provided, combining like terms was used after clearing fractions, which simplified the equation from multiple terms to a single term in terms of 'c'.

Solving for a Variable

Solving for a variable involves isolating it on one side of the equation to find its value.

**Steps to Solve for a Variable**:

• First, clear any fractions or decimals if they complicate calculations.
• Next, simplify both sides of the equation by combining like terms.
• Move all terms involving the variable to one side and constant terms to the opposite side, if needed.
• Finally, isolate the variable by performing inverse operations such as addition, subtraction, multiplication, or division.

In our exercise, solving for 'c' involved dividing both sides of the equation by 13. This isolates 'c', resulting in the solution.