Problem 13 - Algebra 2

Question

Solve each equation. Check each solution. $$ \frac{3 x-2}{12}-\frac{1}{6}=\frac{1}{6} $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 4/3\).

Step 1: Get rid of the fractions

Multiplying every term by 12 (which is the common denominator), we simplify the equation into: \(2(3x - 2) - 2 = 2\)

Step 2: Distribute and Simplify

Distribute the 2 into the bracket and simplify the like terms:\(6x - 4 - 2 = 2\) which further simplifies to \(6x - 6 = 2\)

Step 3: Solve for x

To solve for \(x\), first add 6 to both sides of the equation: \(6x = 8\). Dividing both sides of the equation by 6, we find \(x = 8/6\) which simplifies to \(x = 4/3 or 1.33\)

Step 4: Verify the solution

Substitute \(x = 4/3\) into the original equation.\[\frac{3(4/3)-2}{12}-\frac{1}{6}=\frac{1}{6}\]After simplification, \[\frac{4-2}{12}-\frac{1}{6}=\frac{1}{6}\]This simplifies to:\[\frac{1}{6}-\frac{1}{6}=\frac{1}{6}\]Or \(0 = 0\), which verifies that our solution, \(x = 4/3\), is correct.

Key Concepts

Fractions in Equations

When working with equations that contain fractions, it can seem daunting at first. The key is to simplify the equation by eliminating the fractions. This is typically achieved by finding a common denominator for all terms involved. In our exercise, the denominators were simplified by multiplying each term by 12, which is the least common multiple of the denominators 6 and 12. This step is crucial because once the fractions are gone, you are left with a much simpler equation.
For example, consider the equation with fractions:

• \( \frac{3x - 2}{12} - \frac{1}{6} = \frac{1}{6} \)

By multiplying each side by 12, you eliminate the fractions, resulting in simpler terms to work with like \( 2(3x - 2) - 2 = 2 \).
This process of eliminating fractions not only makes your algebraic manipulations easier but also minimizes potential errors in calculations.

Algebraic Manipulation

Once you've eliminated fractions from your equation, you're left with a more straightforward algebraic problem. Algebraic manipulation involves rearranging and simplifying the equation to find the solution for the variable.
Taking our example further, after clearing fractions, we had the equation:

• \( 2(3x - 2) - 2 = 2 \)

The next step is to "distribute" or apply the distributive property, which means you distribute the 2 into the parenthesis result in \( 6x - 4 \). Remember that whenever you distribute, you're multiplying every term inside the parenthesis by the term outside.
Following distribution, you simplify by combining like terms if possible. Here, you subtract 6 from both sides, resulting in a simplified equation: \( 6x = 8 \).
To find \( x \), divide both sides by 6. This brings us to the solution: \( x = \frac{4}{3} \). Effective algebraic manipulation allows you to isolate and solve for your variable with clarity and precision.

Verifying Solutions

Verifying your solution means checking that your value for \( x \) satisfies the original equation. It's a crucial step to ensure there are no errors in your calculations and that your solution is correct.
In this exercise, after finding \( x = \frac{4}{3} \), you return to the original equation:

• \( \frac{3x - 2}{12} - \frac{1}{6} = \frac{1}{6} \)

Substitute \( \frac{4}{3} \) for \( x \) into the equation and simplify both sides:

• \( \frac{3(4/3) - 2}{12} - \frac{1}{6} = \frac{1}{6} \)

After substituting, simplify to see if both sides are equal.If both sides equal, it confirms your solution \( x = \frac{4}{3} \) checks out. When both sides balance, like in this case \( 0 = 0 \), it reassures you that your solution is correct, giving you confidence in your understanding of solving equations.