Problem 15 - Precalculus

Question

Solve each equation. $$\frac{1}{x}=\frac{4}{x}-1$$

Step-by-Step Solution

Answer

The solution is \( x = 3 \).

Step 1: Understand the Equation

The equation given is \( \frac{1}{x} = \frac{4}{x} - 1 \). Our goal is to find the value of \( x \) that satisfies this equation.

Step 2: Isolate Fractions

Since there are fractions involved, we need to manipulate the equation to make it easier to solve. We do that by subtracting \( \frac{4}{x} \) from both sides of the equation:\[ \frac{1}{x} - \frac{4}{x} = -1 \]

Step 3: Simplify the Left Side

Combine the fractions on the left side of the equation by subtracting them:\[ \frac{1 - 4}{x} = \frac{-3}{x} \]So, the equation becomes \( \frac{-3}{x} = -1 \).

Step 4: Eliminate the Fraction

To eliminate the fraction, multiply both sides by \( x \) to clear the denominator:\[ -3 = -1x \]

Step 5: Solve for x

Divide both sides by -1 to solve for \( x \):\[ x = 3 \]

Step 6: Verify the Solution

Plug \( x = 3 \) back into the original equation to verify it's correct:\[ \frac{1}{3} = \frac{4}{3} - 1 \] This simplifies to \( \frac{1}{3} = \frac{1}{3} \), confirming the solution is accurate.

Key Concepts

Fractions

Fractions are a way of expressing numbers that are not whole. They represent parts of a whole and are written as two numbers separated by a line, also known as a fraction bar. The number on top is called the numerator, and the number on the bottom is the denominator. For example, in the fraction \( \frac{1}{x} \), 1 is the numerator, and \( x \) is the denominator.
Understanding fractions is important as they often appear in equations and require careful handling. When dealing with equations, especially those involving fractions, it's crucial to manage the denominators properly.

• To combine fractions, ensure they have a common denominator.
• Subtraction of fractions involves lining up like terms over a common denominator.

In our equation \( \frac{1}{x} = \frac{4}{x} - 1 \), fractions appear on both sides. Our task first involves managing these fractions to simplify the equation and solve for \( x \).

Isolate Variable

Isolating the variable in an equation means rearranging the equation so that the unknown variable is on one side and all other terms are on the other side. This helps in deducing the value of the variable easily.
When isolating a variable, it's imperative to perform the same operation on both sides of the equation to maintain balance.
In the given equation \( \frac{1}{x} = \frac{4}{x} - 1 \), we start by eliminating the \( \frac{4}{x} \) term from the right-hand side. We substract this term from both sides:
\[ \frac{1}{x} - \frac{4}{x} = -1 \]
After this operation, the equation is transformed, allowing us to isolate and eventually solve for \( x \).

Equation Manipulation

Equation manipulation involves changing the appearance of an equation without altering its solutions. This can include operations like addition, subtraction, multiplication, and elimination of fractions.
A major step in this process is to simplify the equation by working with like terms and simplifying expressions.

• Combine fractions or terms that have a common denominator or are similar in form.
• Multiply or divide both sides of the equation, if necessary, to clear fractions.

In our specific case, we used subtraction to manipulate parts of the equation \( \frac{1}{x} - \frac{4}{x} = -1 \). After simplification, the equation looks like \( \frac{-3}{x} = -1 \). Multiplying both sides by \( x \) removes the denominator, assisting further manipulation in solving for \( x \).

Verify Solution

Verifying a solution ensures that the value obtained for the variable satisfies the original equation. This step confirms whether the solution is correct.
To verify, substitute the calculated value back into the original equation and check if both sides of the equation are equal.
In our problem, after solving for \( x \), we found \( x = 3 \). By substituting \( 3 \) back into the original equation, we have:
\[ \frac{1}{3} = \frac{4}{3} - 1 \]
Simplifying the right-hand side gives \( \frac{1}{3} \), which matches the left-hand side. This confirms that our solution, \( x = 3 \), is accurate and satisfies the equation.
Verifying solutions is a crucial habit in algebra to avoid errors and ensure reliability in solving equations.