Problem 49 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. \(39-56:\) Find the solutions of the equation. $$ 27 x^{3}=(x+5)^{3} $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{5}{2}\).

Step 1: Simplify the Equation

The given equation is \( 27x^3 = (x+5)^3 \). Notice that both sides of the equation are perfect cubes. We can write the equation as \((3x)^3 = (x+5)^3\).

Step 2: Use Cube Root Property

Since both sides of the equation are cubes, we can take the cube root of both sides to simplify it. This gives us \(3x = x + 5\).

Step 3: Rearrange the Equation

Rearrange the equation to isolate \(x\) on one side: \(3x - x = 5\), which simplifies to \(2x = 5\).

Step 4: Solve for x

Divide both sides of the equation by 2 to solve for \(x\): \(x = \frac{5}{2}\).

Step 5: Verify the Solution

Substitute \(x = \frac{5}{2}\) back into the original equation to ensure it satisfies the equation: \(27(\frac{5}{2})^3 = (\frac{5}{2} + 5)^3\). Simplifying both sides confirms the equality holds, verifying the solution.

Key Concepts

Cube Root Property

Understanding the cube root property can greatly simplify solving equations where both sides are cubes. Imagine a cube of sugar you want to break perfectly into smaller blocks. The cube root is basically a question of what number multiplied by itself three times will give the original number. In the case of algebra, when you have an equation like

• \(a^3 = b^3\),

this property allows us to take the cube root of both sides to get \(a = b\). This is crucial because it turns a complex cubic equation into a much simpler linear equation. Like with our sugar cube, once it's broken down, you can handle the pieces much more easily.

Equation Solving

Solving equations is like untangling a ball of yarn to find its start. When faced with an equation such as

• \(3x = x + 5\),

the goal is to find the value of \(x\) that makes the equation true. We start by isolating the variable \(x\) on one side of the equation. This often involves moving terms from one side of the equation to the other.
In this specific problem, we moved \(x\) from the right side to the left by performing the subtraction \(3x - x\). This is done by assuming basic properties of equality, similar to balancing a scale. Whatever you do to one side, you must do to the other. This leads us to \(2x = 5\). Once the equation is simplified, finding \(x\) becomes straightforward: divide both sides by 2, leading to \(x = \frac{5}{2}\). Always remember, solving equations is a logical process. Solve step by step, and you will find the solution.

Algebraic Manipulation

Algebraic manipulation is a powerful tool allowing one to reshape equations and expressions to understand or solve them. Consider algebra like a puzzle where pieces (numbers and variables) need to fit perfectly to reveal a solution. In our example, we started with:

• \(27x^3 = (x+5)^3\).

Manipulation involves recognizing patterns, such as both sides being perfect cubes. By identifying and applying properties like cube roots, we simplify complex equations. Rearranging terms (

• from \(3x = x + 5\) to \(2x = 5\)

) is another form of manipulation, enabling easier solution finding. This process requires changing the equation's appearance without altering its fundamental truth. Once simplified, we employ basic division to solve for \(x\). Through strategic operations, algebraic manipulations make complicated expressions solvable, much like rearranging puzzling pieces until the picture is clear.