Problem 74 - Understanding Elementary Algebra with Geometry

Question

Solve for \(x: \frac{x}{3}-\frac{x-1}{4}=\frac{x+1}{2}\)

Step-by-Step Solution

Answer

The solution is \( x = -\frac{3}{5} \).

Step 1: Find a common denominator

The denominators in the equation are 3, 4, and 2. The least common multiple (LCM) of these denominators is 12. So, rewrite each term with a denominator of 12.

Step 2: Rewrite each fraction

Convert each term so the denominator is 12:\[\frac{x}{3} = \frac{4x}{12}, \frac{x-1}{4} = \frac{3(x-1)}{12}, \frac{x+1}{2} = \frac{6(x+1)}{12}\]

Step 3: Multiply through by the common denominator

Multiply every term in the equation by 12 to clear the denominators:\[12 \times \frac{4x}{12} - 12 \times \frac{3(x-1)}{12} = 12 \times \frac{6(x+1)}{12}\]which simplifies to:\[4x - 3(x-1) = 6(x+1)\]

Step 4: Simplify the equation

Distribute the terms within the equation:\[4x - 3x + 3 = 6x + 6\] which simplifies to:\[x + 3 = 6x + 6\]

Step 5: Isolate the variable

Move the terms involving x to one side and constant terms to the other:\[x - 6x = 6 - 3\]This simplifies to:\[-5x = 3\]

Step 6: Solve for x

Divide both sides by -5 to isolate x:\[x = -\frac{3}{5}\]

Key Concepts

Finding Common Denominators

The first step in solving the equation \(\frac{x}{3} - \frac{x-1}{4} = \frac{x+1}{2}\) is to find a common denominator for all the fractions. A common denominator is a shared multiple of the denominators of all fractions involved.

Here’s how we do it:

• Identify the denominators: 3, 4, and 2.
• Find the least common multiple (LCM) of these denominators. The LCM of 3, 4, and 2 is 12.
• Rewrite each fraction so that they all have the common denominator 12.

So, the fractions are rewritten as:

\[\begin{equation} \frac{x}{3} = \frac{4x}{12}, \frac{x-1}{4} = \frac{3(x-1)}{12}, \frac{x+1}{2} = \frac{6(x+1)}{12} \end{equation}\]

This makes it easier to manage the equation, as they now share the same denominator.

Simplifying Algebraic Expressions

Simplifying algebraic expressions means making them simpler without changing their value. It can involve combining like terms or applying arithmetic operations.

After finding common denominators, we multiply through by 12 to clear the fractions:

\[\begin{equation} 12 \times \frac{4x}{12} - 12 \times \frac{3(x-1)}{12} = 12 \times \frac{6(x+1)}{12} \end{equation}\]

This gives:

\[\begin{equation} 4x - 3(x-1) = 6(x+1) \end{equation}\]

• Distribute the terms within the equation to simplify:
• \[\begin{equation} 4x - 3x + 3 = 6x + 6 \end{equation}\]

Now, the equation looks simpler and can be further simplified to:

\[\begin{equation} x + 3 = 6x + 6 \end{equation}\]

By distributing and combining like terms, we make the equation much easier to handle.

Isolating the Variable

Isolating the variable means getting the variable (in this case, x) by itself on one side of the equation.

Let's start with the simplified equation:

\[\begin{equation} x + 3 = 6x + 6 \end{equation}\]

We'll move all x terms to one side and constant terms to the other:

\[\begin{equation} x - 6x = 6 - 3 \end{equation}\]

Combining like terms, we get:

\[\begin{equation} -5x = 3 \end{equation}\]

Finally, divide both sides by -5 to solve for x:

\[\begin{equation} x = -\frac{3}{5} \end{equation}\]

So, the solution is x = -\(\frac{3}{5}\). We have successfully isolated x and found its value.