Problem 86 - Algebra 2

Question

Solve each equation. \(x+\frac{3}{16}=-\frac{1}{2}\)

Step-by-Step Solution

Answer

x = -\frac{11}{16}

Step 1: Isolate the variable

To solve for \(x\), we need to move \(\frac{3}{16}\) to the right side of the equation. We do this by subtracting \(\frac{3}{16}\) from both sides. The equation becomes: \(x = -\frac{1}{2} - \frac{3}{16}\).

Step 2: Find a common denominator

Before we can subtract \(-\frac{1}{2}\) and \(\frac{3}{16}\), we need to have a common denominator. Since 16 is a multiple of 2, we convert \(-\frac{1}{2}\) to \(-\frac{8}{16}\).

Step 3: Perform the subtraction

Now that both fractions have the same denominator, we subtract: \(-\frac{8}{16} - \frac{3}{16} = -\frac{11}{16}\).

Step 4: Write the final solution

The final answer for \(x\) is \(x = -\frac{11}{16}\).

Key Concepts

Fractions

Fractions represent a part of a whole and are written as one number over another, separated by a line, such as \(\frac{3}{16}\). The top number is called the numerator, and it tells us how many parts we have. The bottom number is the denominator, showing how many parts make up a whole.
Understanding fractions is crucial when solving equations because we often need to perform operations with them, such as adding or subtracting.

• When fractions have the same denominator, you can easily add or subtract the numerators while keeping the denominator unchanged.
• If fractions have different denominators, you'll need a common denominator before you can add or subtract them.

In the given exercise, we needed to perform a subtraction operation with fractions, highlighting the need for a common denominator.

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions, allowing you to add or subtract them. The denominator acts as a 'common ground,' making calculations easier. In other words, when fractions share a common denominator, their numerators can be directly compared or calculated.
To find a common denominator, you typically look for the least common multiple (LCM) of the denominators. In the exercise, the fractions \(-\frac{1}{2}\) and \(\frac{3}{16}\) required a common denominator for subtraction.

• If you have fractions like \(-\frac{1}{2}\) and \(\frac{3}{16}\), you can choose the smallest number both denominators divide into without remainder, like 16 in this case.
• Convert both fractions to have this new common denominator, transforming \(-\frac{1}{2}\) into \(-\frac{8}{16}\).

This method clears the way for straightforward subtraction, essential in obtaining the final solution.

Variable Isolation

Variable isolation is the process of getting the variable alone on one side of the equation to find its value. This means you have a statement like \(x = ...\) that tells you exactly what the variable equals. It's an important step in solving any equation.
In our exercise, the variable \(x\) initially appears with other terms. To isolate \(x\), we need to move all other terms to the opposite side of the equation.

• This often involves using inverse operations, such as addition versus subtraction or multiplication versus division.
• In the given exercise, we subtracted \(\frac{3}{16}\) from both sides to shift it away from \(x\), leading to \(x = -\frac{1}{2} - \frac{3}{16}\).

Once this step is completed with the common denominator, you simply compute the arithmetic operation to find \(x\). This technique effectively unveils the value of the desired variable, in our case, \(x = -\frac{11}{16}\).