Problem 32 - Algebra for College Students

Question

Solve each equation. \(\frac{x-2}{5}-\frac{x-3}{4}=-\frac{1}{20}\)

Step-by-Step Solution

Answer

The solution to the equation is \( x = 8 \).

Step 1: Eliminate the Fractions

The given equation is \( \frac{x-2}{5}-\frac{x-3}{4}=-\frac{1}{20} \). Start by finding the least common denominator (LCD) to eliminate the fractions. The denominators are 5, 4, and 20. The LCD is 20. Multiply every term in the equation by 20 to clear the fractions:\[ 20 \times \frac{x-2}{5} - 20 \times \frac{x-3}{4} = 20 \times -\frac{1}{20} \]This simplifies to:\[ 4(x-2) - 5(x-3) = -1 \].

Step 2: Distribute and Simplify

Distribute the numbers across the terms inside the parentheses in the equation obtained:\[ 4(x-2) - 5(x-3) = -1 \]This gives:\[ 4x - 8 - 5x + 15 = -1 \].Now combine like terms:\[ (4x - 5x) + (-8 + 15) = -1 \]\[ -x + 7 = -1 \].

Step 3: Solve for the Variable

Now, solve for \( x \) in the equation \( -x + 7 = -1 \). Start by isolating \( x \) on one side. Subtract 7 from both sides:\[ -x + 7 - 7 = -1 - 7 \]\[ -x = -8 \].Multiply by -1 to solve for \( x \):\[ x = 8 \].

Key Concepts

Understanding the Least Common Denominator (LCD)

When dealing with equations that include fractions, it's often helpful to eliminate the fractions by using the least common denominator (LCD). This process can make the equation easier to solve. In the example equation \( \frac{x-2}{5}-\frac{x-3}{4}=-\frac{1}{20} \), the denominators are 5, 4, and 20.

• The first step is finding the LCD of these numbers. The LCD is the smallest number that each of the denominators can divide into without leaving a remainder.
• For the numbers 5, 4, and 20, the LCD is 20, since 20 is a multiple of all three numbers.
• Once the LCD is determined, each term in the equation is multiplied by the LCD. This allows the fractions to "cancel out."

After multiplying through by the LCD, our equation becomes: \[ 4(x-2) - 5(x-3) = -1 \] Using the LCD streamlines the equation by converting it into an expression without fractions.

Applying the Distributive Property

The distributive property is a crucial tool for simplifying expressions, especially when dealing with equations like \( 4(x-2) - 5(x-3) = -1 \).

• This property allows you to multiply a single term by each term within a parenthesis. It uses the rule: \( a(b+c) = ab + ac \).
• In our equation, applying the distributive property gives us: \( 4(x-2) = 4x - 8 \) and \( -5(x-3) = -5x + 15 \).

After distributing, the equation becomes:\[ 4x - 8 - 5x + 15 = -1 \] This step ensures each term is taken into consideration, setting the stage for combining like terms in the next section.

Combining Like Terms in an Equation

Combining like terms is another fundamental concept that helps simplify expressions and equations. It involves grouping and combining terms with the same variable.

• In our equation \( 4x - 8 - 5x + 15 = -1 \), the like terms are \( 4x \) and \(-5x\), as they both include the variable \( x \).
• Similarly, the constants \( -8 \) and \( 15 \) are also like terms.
• By bringing together the like terms, we simplify our equation: \( (4x - 5x) + (-8 + 15) = -1 \), which simplifies further to \( -x + 7 = -1 \).

This combination of terms simplifies the equation so that solving for the variable becomes more straightforward. Removing complexities from the equation allows us to quickly isolate and solve for \( x \). In our final simplified equation, solving for \( x \) becomes an easy task, arriving at \( x = 8 \).