Problem 64 - Elementary Algebra Within Reach

Question

In Exercises \(63-66\), solve the equation. $$ \frac{9}{x+3}=15 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -\frac{12}{5}\)

Step 1: Rewrite the equation

In order to make this equation easier to solve, one can start by getting rid of the fraction. This is done by multiplying both sides by (x+3): \((x + 3)\frac{9}{x + 3}=15(x + 3)\). Simplifying, we get \(9 = 15x + 45\)

Step 2: Isolate the variable

We can now isolate the variable x by subtracting 45 from both sides of the equation. This results in the equation -36 = 15x.

Step 3: Solve for x

You can now find the value of x by dividing both sides of the equation by 15. This results in \(x = -\frac{36}{15}\). Reduce the fraction to its simplest form to get \(x = -\frac{12}{5}\)

Key Concepts

Fraction Simplification

When you deal with fractions, sometimes they can look daunting, but they're not too bad when you break them down. In solving rational equations, simplifying fractions is often a key step. Here’s why it's important:

• Fraction simplification untangles the equation and makes calculations easier.
• A fraction is in its simplest form when the numerator and denominator have no common factors other than 1.
• Simplifying may help reveal the most straightforward path to solve an equation.

To simplify a fraction like \(-\frac{36}{15}\), find the greatest common divisor (GCD) of the numerator and denominator. Here, both 36 and 15 are divisible by 3. This reduces our fraction to \(-\frac{12}{5}\). This clearer form is easier to interpret and use in further calculations.

Isolating Variables

A crucial skill in algebra is isolating the variable. This means rearranging the equation so that you have just the variable on one side, typically the left side. This is how you find 'x' or whatever your unknown is:

• Move all terms with the variable to one side of the equation.
• Shift constant terms to the opposite side.
• Perform inverse operations to solve for the variable.

In our example, after multiplying both sides of the equation and simplifying, the equation becomes \(9 = 15x + 45\). By subtracting 45 from both sides to get \(-36 = 15x\), the variable x is isolated, allowing us to solve for its value.

Algebraic Equations

Algebraic equations are statements of equality containing unknowns and variables. Solving such equations often involves several strategic steps:

• Understand the equation and what it represents.
• Perform operations to simplify and rearrange terms as needed.
• Check for simplification opportunities to make life easier.

In the case of \(\frac{9}{x+3} = 15\), by understanding the roles of numerator and denominator, you can effectively clear the fraction early on. By addressing equations systematically, you ensure a clear path to find the unknown variable.

Multiplying Both Sides of an Equation

To solve equations that include fractions, a common technique is to eliminate the fraction by multiplying both sides by the same term. This doesn't change the equation's balance, as multiplication affects both sides equally:

• Identify a term that, when multiplied, will simplify the equation.
• Apply this term to both sides to ensure the equality still holds.
• Simplify after multiplying to keep the equation tidy and understandable.

For instance, multiplying both sides of \(\frac{9}{x+3} = 15\) by \(x+3\) cancels out the fraction, resulting in a linear equation \(9 = 15x + 45\). This step is crucial for simplifying and preparing the equation for solving.