Problem 18 - Elementary Algebra

Question

Solve. $$ (15 x-3) 2=0 $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{1}{5}\).

Step 1: Distribute the Factor

Distribute the 2 across the equation to simplify it. Multiply each term inside the parentheses by 2: \[(15x - 3) \times 2 = 30x - 6\]

Step 2: Set the Equation to Zero

With the equation simplified, ensure the equation is set to zero: \[30x - 6 = 0\]

Step 3: Isolate the Variable

Add 6 to both sides to begin isolating the variable \(x\):\[30x - 6 + 6 = 0 + 6\]This simplifies to:\[30x = 6\]

Step 4: Solve for the Variable

Divide both sides by 30 to solve for \(x\): \[x = \frac{6}{30}\]Simplify the fraction to: \[x = \frac{1}{5}\]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps to simplify expressions and solve equations. It involves distributing a factor outside parentheses to each term inside. Consider the equation

• (15x - 3) 2 = 0

To apply the distributive property here, multiply each term within the parentheses by 2:

• 15x times 2 becomes 30x
• -3 times 2 becomes -6

After distributing, the equation simplifies to 30x - 6 = 0. This step is crucial because it turns a complex expression into a simpler linear equation, which makes it easier to handle in subsequent steps.
Remember, distributing is an essential tool for breaking down complex, seemingly unsolvable equations into simpler components. It's a handy way to manage terms and make calculations straightforward.

Isolating Variables

Isolating the variable, typically represented as "x" in algebra, focuses on getting the variable alone on one side of the equation. Following the equation after distribution, we have 30x - 6 = 0. Our goal is to make the equation look like x = something. To isolate x, start by removing the number directly connected to it (in this case, -6). Add 6 to both sides:

• 30x - 6 + 6 = 0 + 6
• This simplifies the equation to 30x = 6

Every action taken on one side must be mirrored on the other to maintain the equation's balance. Therefore, adding 6 eliminates the subtraction, winnowing down the equation and drawing us closer to finding the variable's value. Isolating variables is like peeling layers from an onion—step by step until you're left with what's essential.

Solving Equations

Once the variable is isolated, the next step is to solve the equation. The equation at this stage is 30x = 6. Solving for x involves determining its value that satisfies the equation. Here, you need to divide both sides by 30 to find x:

• x = \( \frac{6}{30} \)

By dividing, you're effectively undoing the multiplication present in the equation. Solving equations is the ultimate goal in algebra, revealing the value of the unknown or variable. This step verifies all previous work and concludes the problem, showing that the methods and properties applied hold true.

Simplifying Fractions

Simplifying fractions entails reducing them to their simplest form, ensuring the numerator and denominator have no common factors other than 1. In our equation, after solving for x, we're left with the fraction \( \frac{6}{30} \). To simplify:

• Identify common factors of 6 and 30, which is 6
• Divide both the numerator and the denominator by 6
• \( \frac{6}{6} = 1 \) and \( \frac{30}{6} = 5 \)

Thus, \( \frac{6}{30} \) simplifies to \( \frac{1}{5} \). Simplification is crucial for clarity, making it easier to interpret results and uncover underlying relationships. Understanding how fractions work will allow you to handle them better in any algebraic scenario, ensuring you're left with the neatest, clearest expression possible.