Problem 73 - Algebra A Combined Function

Question

\(-x-\frac{4}{5}=x+\frac{1}{2}+\frac{2}{5}\)

Step-by-Step Solution

Answer

The solution is \(x = -\frac{17}{20}\).

Step 1: Combine Like Terms

The given equation is \(-x-\frac{4}{5}=x+\frac{1}{2}+\frac{2}{5}\). First, we need to combine the terms on the right side that are simply numbers: \(\frac{1}{2}\) and \(\frac{2}{5}\). To do that, find a common denominator (which is 10) and combine the fractions: \(\frac{1}{2} = \frac{5}{10}\) and \(\frac{2}{5} = \frac{4}{10}\). So, the right side becomes \(x + \frac{9}{10}\). Now the equation is \(-x-\frac{4}{5} = x + \frac{9}{10}\).

Step 2: Move Variable Terms to One Side

To solve for \(x\), we move all terms involving \(x\) to one side of the equation. Add \(x\) to both sides to eliminate \(-x\) on the left side: \(-x - \frac{4}{5} + x = x + \frac{9}{10} + x\). This simplifies to \(-\frac{4}{5} = 2x + \frac{9}{10}\).

Step 3: Isolate the Variable Term

Subtract \(\frac{9}{10}\) from both sides to isolate the variable term \(2x\): \(-\frac{4}{5} - \frac{9}{10} = 2x\). Convert \(-\frac{4}{5}\) to units of 10, giving \(-\frac{8}{10}\). Now calculate: \(-\frac{8}{10} - \frac{9}{10} = -\frac{17}{10}\). So, \(-\frac{17}{10} = 2x\).

Step 4: Solve for the Variable

Divide both sides by 2 to solve for \(x\): \(-\frac{17}{10} \div 2 = x\). Dividing by 2 is the same as multiplying by \(\frac{1}{2}\), so we get \(x = -\frac{17}{20}\).

Key Concepts

Combining Like Terms

In mathematical equations, combining like terms is a fundamental step to simplify expressions and make the equation easier to manage. Like terms are terms that contain the same variable raised to the same power. They differ only by their coefficients. For instance, in the exercise equation \(-x - \frac{4}{5} = x + \frac{1}{2} + \frac{2}{5}\), the like terms on the left and right side are those without any variables attached.
To combine the like terms \(\frac{1}{2} + \frac{2}{5}\) on the right side, we need to operate on these fractions and find a way to add them together, leading us to the next concept: finding common denominators to add fractions effectively.

Common Denominators

Calculating a common denominator is essential when adding or subtracting fractions. The denominator is the bottom number of a fraction that shows how many equal parts the whole is divided into. In our exercise, we need to combine \(\frac{1}{2}\) and \(\frac{2}{5}\). To do this, we look for a common denominator that both 2 and 5 can divide evenly, which is 10 in this case.

• Convert each fraction to an equivalent fraction with the denominator 10.
• \(\frac{1}{2}\) becomes \(\frac{5}{10}\) because \(1 \times 5 = 5\) and \(2 \times 5 = 10\).
• \(\frac{2}{5}\) becomes \(\frac{4}{10}\) because \(2 \times 2 = 4\) and \(5 \times 2 = 10\).

Now, we can easily add these two equivalent fractions: \(\frac{5}{10} + \frac{4}{10} = \frac{9}{10}\). This way, we simplify the right side of the equation to \(x + \frac{9}{10}\), thus combining like terms successfully.

Isolating the Variable

Isolating the variable is a crucial step in solving equations, especially linear equations. To isolate a variable means to get the variable by itself on one side of the equation. This allows you to see exactly what the variable equals. In our equation, after combining like terms, we have \(-x - \frac{4}{5} = x + \frac{9}{10}\).

To start isolating \(x\), you move all terms involving \(x\) to one side. This is done by adding \(x\) to both sides, which helps eliminate \(-x\) on the left side: \(-x - \frac{4}{5} + x = x + \frac{9}{10} + x\), simplifying to \(-\frac{4}{5} = 2x + \frac{9}{10}\).
Next, subtract \(\frac{9}{10}\) from both sides to further isolate the term \(2x\): \(-\frac{4}{5} - \frac{9}{10} = 2x\). Converting \(-\frac{4}{5}\) to a denominator of 10, we get \(-\frac{8}{10}\). Then, subtract: \(-\frac{8}{10} - \frac{9}{10} = -\frac{17}{10}\). Thus, the equation becomes \(-\frac{17}{10} = 2x\).
The final step is to solve for \(x\) by dividing both sides by 2, which gives \(x = -\frac{17}{20}\). Each of these steps helps break down the equation, leading us to the solution.