Problem 115 - Introductory Algebra for College Students

Question

Solve: \(8 x-\frac{x}{6}=\frac{1}{6}-8\)

Step-by-Step Solution

Answer

The solution to the equation \(8x - \frac{x}{6} = \frac{1}{6} - 8\) is \(x = -1\).

Step 1: Combine Like Terms

One has to begin by first combining like terms on each side of the equation. This provides a simplified equation that is easier to work with. The left side of the equation is \(8x - \frac{x}{6}\). Multiply \(x\) by \(\frac{8}{6}\) to combine the like terms resulting in \(\frac{48x}{6} - \frac{x}{6} = \frac{47x}{6}\). As there exist no like terms on the right side, it stays as \(\frac{1}{6} - 8\). Hence the simplified equation is \(\frac{47x}{6} = \frac{1}{6} - 8\).

Step 2: Simplify Right Side

Next step is to simplify the right side. Before that convert 8 to a fraction with 6 as denominator: \(8 = \frac{48}{6}\). Therefore, the right side \(\frac{1}{6} - 8\) becomes \(\frac{1}{6} - \frac{48}{6}\) equals to \(-\frac{47}{6}\).

Step 3: Isolate x

Since \(\frac{47x}{6} = -\frac{47}{6}\), we need to isolate \(x\). To do this, multiply both sides by \(\frac{6}{47}\). This gives us \(x = -\frac{47}{6} \times \frac{6}{47}\) which simplifies to \(x = -1\).

Key Concepts

Combining Like Terms

The process of combining like terms is a crucial first step in solving linear equations. Like terms are terms that have the same variables raised to the same power. For instance, in the equation from the exercise, we have two terms that contain the variable 'x': \(8x\) and \(-\frac{x}{6}\). When we combine like terms, we streamline the equation, making it more manageable.

When we look at our equation, we can see that to combine the terms containing 'x,' we need a common denominator. Here, the denominator is 6. Multiplying 8 by 6 gives us \( \frac{48x}{6} \), which allows us to combine with \(-\frac{x}{6}\) to get \(\frac{47x}{6}\). This is a significant step, because it simplifies the left side of the equation into a single term that we can then easily manipulate.

Simplifying Fractions

Simplifying fractions is another foundational skill that comes into play when solving equations. A fraction is simplified when the numerator and the denominator have no common factors other than 1. In the context of our exercise, we encounter the term \(-\frac{47}{6}\), which is already simplified since 47 and 6 do not share any common factors.

However, before we got to that simplified fraction, we had to deal with converting a whole number to a fraction, which in this case was converting 8 to \(\frac{48}{6}\). This enables us to perform subtraction on the right side of the equation with the fraction \(\frac{1}{6}\), resulting in the simplified form of \(-\frac{47}{6}\). The process of simplifying right here allows for a clear, direct comparison between the two sides of the equation.

Isolating Variables

Isolating the variable is the final and decisive step in solving a linear equation. Once we have simplified our equation as much as possible, we want to get the variable 'x' by itself—that is, to isolate it—so we can find its value. In our example, isolating 'x' involves getting rid of the fraction \(\frac{47}{6}\) that is multiplied by it.

To isolate 'x', we perform the inverse operation. Initially, 'x' is being multiplied by \(\frac{47}{6}\), so we multiply both sides of the equation by the inverse of this coefficient, which is \(\frac{6}{47}\). By doing so, the multiplication by \(\frac{47}{6}\) on the left side is undone, and on the right side, we end up multiplying \(-\frac{47}{6}\) by \(\frac{6}{47}\), which simplifies to -1. Consequently, we have successfully isolated 'x', finding that \(x = -1\), which is the solution to the equation.