Problem 15 - Prealgebra and Introductory Algebra

Question

Solve. $$\frac{12}{3 x-2}=3$$

Step-by-Step Solution

Answer

The solution to the equation \( \frac{12}{3x - 2} = 3 \) is \( x = 2 \).

Step 1: Isolate the Fraction

The first step is to isolate the fraction by multiplying both sides of the equation with \(3x - 2\). Our equation given is: \( \frac{12}{3x - 2} = 3 \). When we multiply both sides by \(3x - 2\), we get: \( 12 = 3 * (3x - 2) \).

Step 2: Expand the Right Side

Next, we simplify the right side of the equation by expanding \(3 * (3x - 2)\), which gives us: \( 12 = 9x - 6 \)

Step 3: Solve for x

We can now solve this simple linear equation for x by first adding 6 to both sides: \(12 + 6 = 9x - 6 + 6 \) Thus, we get \(18 = 9x\). Then, simply divide both sides by 9 to solve for x: \( \frac{18}{9} = x \) or \( x = 2 \).

Key Concepts

Isolate the Variable

One of the critical steps in solving linear equations is isolating the variable, which means separating the variable on one side of the equation from the numeric terms. This makes it easier to find its value. In the given exercise, the equation is \( \frac{12}{3x - 2} = 3 \). Here, the variable \( x \) is in the denominator of a fraction. So, our first task is to remove it from there.
To do this, we multiply both sides of the equation by \( 3x - 2 \). This cancels out the denominator on the left, giving us a simpler equation \( 12 = 3 \cdot (3x - 2) \). Now, \( x \) is not trapped within a fraction anymore.

• Understand why multiply: It keeps the equation balanced.
• See the cancellation effect in action: No denominator on the left side.

Now, we have set the stage to perform operations to isolate the variable entirely on one side.

Expand Equations

After isolating the variable within the equation, the next step is expanding equations. Expanding involves removing parentheses and simplifying expressions to make calculations straightforward.
In our example with \( 12 = 3 \cdot (3x - 2) \), expansion means we distribute the \( 3 \) across both terms inside the parentheses. This gives \( 12 = 9x - 6 \).

• Distributive Property: Apply \( a(b + c) = ab + ac \).
• Simplify by distributing: \( 3 \cdot 3x \) becomes \( 9x \).
• Also distribute \( 3 \cdot -2 \), which gives \( -6 \).

By expanding, we convert the equation from its factored form to a more direct form that is easier to manipulate, setting us up for the final solving step.

Linear Equations

Understanding linear equations helps in recognizing these equations and tackling them effectively. A linear equation is an equation of the first degree, meaning it has variables raised to the power of one.
Our example, \( 12 = 9x - 6 \), is in its essence a linear equation. The goal is to find the value of \( x \) that satisfies the equation.

• One variable: Our equation has only one variable, \( x \).
• Straightforward calculation: Add or subtract terms to isolate the variable, then divide to solve for it.
• No exponents or powers: Variables are not squared; keep it simple.

In the final step, you solve for \( x \) by adding \( 6 \) to both sides, yielding \( 18 = 9x \), and then dividing by \( 9 \) to find \( x = 2 \). Linear equations offer a clear pathway to finding solutions through basic arithmetic operations.