Problem 28 - Precalculus Mathematics for Calculus

Question

The given equation is either linear or equivalent to a linear equation. Solve the equation. $$\sqrt{3} x+\sqrt{12}=\frac{x+5}{\sqrt{3}}$$

Step-by-Step Solution

Answer

The solution is \( x = -\frac{1}{2} \).

Step 1: Simplify the Equation

Given the equation \( \sqrt{3} x + \sqrt{12} = \frac{x+5}{\sqrt{3}} \), start by simplifying \( \sqrt{12} \). It can be written as \( 2\sqrt{3} \) because \( \sqrt{12} = \sqrt{4 \times 3} = 2 \sqrt{3} \). Now, the equation becomes \( \sqrt{3} x + 2\sqrt{3} = \frac{x+5}{\sqrt{3}} \).

Step 2: Eliminate the Fraction

Multiply every term in the equation by \( \sqrt{3} \) to eliminate the fraction. So, \( \sqrt{3} \times ( \sqrt{3} x + 2\sqrt{3} ) = \sqrt{3} \times \frac{x+5}{\sqrt{3}} \). This simplifies to \( 3x + 6 = x + 5 \).

Step 3: Rearrange the Equation

Subtract \( x \) from both sides of the equation to get \( 3x - x + 6 = 5 \). Simplifying gives \( 2x + 6 = 5 \).

Step 4: Solve for x

Subtract 6 from both sides to isolate the \( 2x \) term: \( 2x = 5 - 6 \). This simplifies to \( 2x = -1 \).

Step 5: Find the Value of x

Divide both sides by 2 to get \( x = \frac{-1}{2} \). Thus, \( x = -\frac{1}{2} \).

Key Concepts

Simplification

The first step in solving a linear equation often involves simplifying the equation. This process is crucial as it helps you to see the basic structure of the equation by changing it into an equivalent but simpler form.

In our given problem, the simplification begins with addressing radicals or roots. We look at the expression \( \sqrt{12} \). If we break it down, we recognize that 12 can be factored into prime factors: \( 12 = 4 \times 3 \). The square root of each factor gives us \( \sqrt{4} = 2 \) and \( \sqrt{3} \). Therefore, \( \sqrt{12} = 2\sqrt{3} \).

• Identify perfect squares within the radicand.
• Simplify each root by breaking down numbers into their prime factors.

This simplification ensures that the equation becomes more manageable and paves the way for subsequent steps such as fraction elimination and solving the equation.

Equation Solving

Solving an equation means finding the value of the variable that makes the equation true. Once the equation is simplified, it becomes easier to isolate the variable that you are solving for.

In our problem, after eliminating the fraction, we arrive at the equation \( 3x + 6 = x + 5 \). The goal here is to manipulate the equation so that all the terms involving the variable, in this case, \( x \), are on one side of the equation, and all constant numbers are on the other side.

• Subtract \( x \) from both sides to get \( 3x - x + 6 = 5 \).
• Simplify the equation: \( 2x + 6 = 5 \).

By systematically performing these operations, you isolate \( x \) to find its value.

Fraction Elimination

Fractions can make equations look complicated and difficult to solve. Therefore, eliminating fractions often makes the process easier.

In the given problem, the fraction \( \frac{x+5}{\sqrt{3}} \) is present. To eliminate this fraction, you multiply every term in the equation by \( \sqrt{3} \). This operation is done because multiplying by the denominator of the fraction will cancel it out.

• Multiply both sides of the equation by \( \sqrt{3} \): \( \sqrt{3}(\sqrt{3}x + 2\sqrt{3}) = \sqrt{3} \times \frac{x+5}{\sqrt{3}} \).
• This results in \( 3x + 6 = x + 5 \).

By clearing the fraction, the equation not only becomes simpler but also more straightforward for applying standard algebraic methods to find the solution.