Problem 82 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. (Some of the equations have no solution.) $$\frac{8-3 x}{2}-4=\frac{x}{6}$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = \frac{4}{7} \).

Step 1: Simplify The Equation

First, simplify the equation by getting everything in terms of 'x'. We can start by subtracting \( \frac{x}{6} \) from both sides of the equation, and then adding 4 to both sides. This gives us: \( \frac{8-3x}{2} -4 -\frac{x}{6} = 0 \).

Step 2: Combine like terms

On the left hand side, we can multiply the denominators of the fractions out. This gives us: \( 4(8-3x) -2x - 24 = 0 \). Simplifying this yields: \( 32 -12x -2x - 24 = 0 \) which leads to: \( -14x + 8 = 0 \).

Step 3: Solving for 'x'

Next, isolate the variable 'x' by subtracting 8 from both sides of the equation to get: \( -14x = -8 \). Then divide by -14 to solve for 'x'. So, \( x = \frac{8}{14} = \frac{4}{7} \).

Step 4: Check the Solution

Check if the solution is correct by substituting \( x = \frac{4}{7} \) back into the original equation. If both sides of the equation are equal, then our solution is correct.

Key Concepts

Algebraic Fractions

Understanding how to work with algebraic fractions is crucial when solving linear equations. Just like numerical fractions, algebraic fractions involve a numerator and a denominator, but with variables like ‘x’. To simplify an equation with algebraic fractions, you often need to find a common denominator so that you can combine terms or eliminate the fractions entirely. In the given exercise, there’s an initial step where you simplify the equation by addressing the fractions and aiming to get everything in terms of ‘x’. Handling algebraic fractions correctly can help reduce mistakes and make subsequent steps in the problem-solving process smoother.

When you encounter equations such as \( \frac{8-3x}{2} = \frac{x}{6} \), remembering that you can perform the same operation on both sides of the equation to eliminate fractions is essential. This might involve multiplying through by the least common multiple of the denominators, which in this case would be 6, to get rid of the fractions and make the equation easier to solve.

Combining Like Terms

The process of combining like terms is a fundamental step in simplifying algebraic expressions. It involves adding or subtracting terms that have the same variable raised to the same power. In other words, you can only combine terms that have the same literal coefficient. For instance, in the exercise, the terms ‘-3x’ and ‘-x’ are like terms because they both contain the variable ‘x’. Combining them is achieved by adding or subtracting their numerical coefficients: \( -3x - x = -4x \). Recognizing like terms helps in streamlining the equation to its simplest form, which is often required before isolating the variable.

By mastering this skill, you can simplify complex equations and avoid common errors, such as combining unlike terms, which can lead to incorrect solutions. When performing these combinations, always ensure to consider the sign in front of each term, as it will affect the operation you perform.

Isolation of Variables

The goal of isolating the variable is to solve for the unknown in an equation. Isolating the variable means manipulating the equation in a way that gets the variable on one side of the equation and the constants on the other. This is typically done by performing a series of inverse operations, such as adding, subtracting, multiplying, or dividing both sides of the equation, until the variable is left by itself.

In the provided exercise, after combining like terms, you isolate the variable ‘x’ by first subtracting 8 from both sides to remove the constant from the side containing the variable. Doing this provides you with \( -14x = -8 \). Next, you would divide both sides by -14 to find the value of ‘x’. The goal is to ensure the variable stands alone, making it simple to see what ‘x’ equals. Getting comfortable with this process is key to mastering algebra, as it is a fundamental step in solving linear equations.