Problem 87 - Algebra 2

Question

Solve each equation. \(x-\frac{1}{6}=-\frac{7}{9}\)

Step-by-Step Solution

Answer

The solution is \(x = -\frac{11}{18}\).

Step 1: Understand the equation

The given equation is \(x - \frac{1}{6} = -\frac{7}{9}\). We need to solve for \(x\), which means we need to isolate \(x\) on one side of the equation.

Step 2: Add \(\frac{1}{6}\) to both sides

To isolate \(x\), add \(\frac{1}{6}\) to both sides of the equation:\[x - \frac{1}{6} + \frac{1}{6} = -\frac{7}{9} + \frac{1}{6}\] This simplifies to:\[x = -\frac{7}{9} + \frac{1}{6}\]

Step 3: Find a common denominator

To add the fractions \(-\frac{7}{9}\) and \(\frac{1}{6}\), first find a common denominator. The least common multiple of 9 and 6 is 18.

Step 4: Convert fractions to common denominator

Convert \(-\frac{7}{9}\) and \(\frac{1}{6}\) to fractions with denominator 18:\[-\frac{7}{9} = -\frac{14}{18}\]\[\frac{1}{6} = \frac{3}{18}\]

Step 5: Add the fractions

Now add the two fractions with a common denominator:\[-\frac{14}{18} + \frac{3}{18} = -\frac{11}{18}\]

Step 6: Conclusion

The value of \(x\) is \(-\frac{11}{18}\). Therefore, the solution to the equation is:\[x = -\frac{11}{18}\]

Key Concepts

Solving Linear Equations

Linear equations are equations involving an unknown variable raised to the first power. To solve a linear equation, our goal is to isolate this unknown variable on one side of the equation. Let's break it down into manageable steps.

• First, identify the unknown variable you need to solve for. In our example, it's the "x" in the equation.
• The second step is to move all terms with the unknown variable to one side of the equation and constant terms to the other side. This often involves basic operations like addition or subtraction.
• Once the variable is isolated, simplify the equation to find the solution. Remember, whatever operation you apply to one side of the equation, you must also apply to the other to maintain balance.

For example, if you have the equation \(x - \frac{1}{6} = -\frac{7}{9}\), you can isolate \(x\) by adding \(\frac{1}{6}\) to both sides. This step effectively "undoes" the subtraction in the original equation by performing the opposite operation.

Understanding Common Denominators

When working with fractions, especially in equations, finding a common denominator is crucial for operations like addition and subtraction. A common denominator helps you combine fractions easily.

• The denominator is the bottom part of the fraction. It shows how many equal parts the whole is divided into.
• Common denominators are used when two or more fractions need to be compared or combined. Finding a common denominator includes determining the least common multiple (LCM) of the denominators involved.
• Once a common denominator is found, convert each fraction to an equivalent fraction with this new denominator.

For our example, the denominators are 9 and 6. Their LCM is 18. By converting \(-\frac{7}{9}\) to \(-\frac{14}{18}\) and \(\frac{1}{6}\) to \(\frac{3}{18}\), you can easily perform addition or subtraction.

Mastering Fraction Addition

Adding fractions may seem tricky at first, but once you grasp the concept of common denominators, it becomes straightforward. Let's dive into the process.

• First, ensure all fractions in the equation have a common denominator. If they don't, convert them as previously discussed using the least common multiple.
• Once you have a common denominator, add or subtract the numerators—the top parts of the fractions—while keeping the denominator the same.
• The resulting fraction is the solution to your addition problem, but remember to simplify the fraction if possible.

In our example, adding \(-\frac{14}{18}\) and \(\frac{3}{18}\) results in \(-\frac{11}{18}\). The denominator remains 18, and we subtract the numerators \(-14 + 3\) to get \(-11\). It's essential to pay attention to the signs (positive or negative) during this process.