Problem 37 - A Graphical Approach to College Algebra

Question

$$\text { Solve each equation analytically. Check it analytically, and then support your solution graphically.}$$ $$\frac{5}{6} x-2 x+\frac{1}{3}=\frac{1}{3}$$

Step-by-Step Solution

Answer

The solution is \( x = 0 \).

Step 1: Simplify the Equation

Start by combining like terms on the left side of the equation. The equation given is \( \frac{5}{6}x - 2x + \frac{1}{3} = \frac{1}{3} \). Combine \( \frac{5}{6}x \) and \( -2x \). This simplifies to \(-\frac{7}{6}x + \frac{1}{3} = \frac{1}{3} \).

Step 2: Isolate the Variable Term

Subtract \( \frac{1}{3} \) from both sides to isolate the term involving \( x \). This gives \(-\frac{7}{6}x = \frac{1}{3} - \frac{1}{3} \). Simplifying this results in \(-\frac{7}{6}x = 0 \).

Step 3: Solve for \( x \)

To solve for \( x \), divide both sides of the equation by \(-\frac{7}{6} \). This results in \( x = 0 \). Such division confirms that \( x = 0 \) is indeed the solution.

Step 4: Check the Solution

Substitute \( x = 0 \) back into the original equation to verify correctness. The original equation is \( \frac{5}{6}x - 2x + \frac{1}{3} = \frac{1}{3} \). Substitute \( 0 \) for \( x \) to get \( 0 - 0 + \frac{1}{3} = \frac{1}{3} \). This equality is true, confirming our solution is correct.

Step 5: Graphical Support

Graph the expression \( y = \frac{5}{6}x - 2x + \frac{1}{3} \) and \( y = \frac{1}{3} \) as two separate lines on the same coordinate plane. The solution \( x = 0 \) matches the point where these two lines intersect, as both expressions yield \( y = \frac{1}{3} \) when \( x = 0 \).

Key Concepts

Equation Solving

Solving equations analytically involves performing algebraic operations to isolate the variable. In this case, the given equation can seem a bit tricky at first due to the presence of fractions. Let’s break it down step by step.

Here’s what was done in the solution:

• **Combine Like Terms:** Starting with the equation \( \frac{5}{6}x - 2x + \frac{1}{3} = \frac{1}{3} \), the first step is to combine like terms, which are the terms involving \( x \): \( \frac{5}{6}x - 2x \) simplifies to \( -\frac{7}{6}x \).
• **Isolate the Variable Term:** Next, subtract \( \frac{1}{3} \) from both sides to get rid of the constant term on the left. This simplifies the equation to \( -\frac{7}{6}x = 0 \).
• **Solve for \( x \):** Finally, to solve for \( x \), divide both sides by \( -\frac{7}{6} \). This results in \( x = 0 \).

Every subtraction, division, and simplification step leads you closer to finding \( x \). This systematic approach is crucial in solving equations correctly.

Graphical Verification

Graphical verification is a technique used to visually confirm the solution of an equation by plotting graphs. This method not only serves as a confirmation but also helps in understanding how equations behave visually.

In our case, the equation \( \frac{5}{6}x - 2x + \frac{1}{3} = \frac{1}{3} \) was represented by the lines:

• **First line:** \( y = \frac{5}{6}x - 2x + \frac{1}{3} \)
• **Second line:** \( y = \frac{1}{3} \)

By plotting both as lines on a graph, the intersection point reveals the solution for \( x \). In this scenario, both lines intersect at \( x = 0 \), suggesting that the solution obtained analytically is accurate. Graphical verification is a powerful method because it provides a visual confirmation that enhances understanding.

Algebraic Simplification

Algebraic simplification is an essential part of solving equations. It involves reducing an expression to its simplest form to make calculations more straightforward.

### Steps of Simplification

• **Identify Like Terms:** Combine terms that involve the same variables. This is crucial as it reduces the complexity of the expression.
• **Eliminate Fractions:** Adjust the terms involving fractions, such as \( \frac{5}{6}x \) in our example, by common denominators or direct calculations to simplify the equation further.
• **Simplify Constants:** Deal with constant terms independently, as seen with \( \frac{1}{3} \), ensuring their smooth transition from one side of the equation to the other.

Through these steps, the equation became manageable, allowing precise isolation and solving for \( x \). Simplification not only eases calculation but also clarifies the equation's structure, which is beneficial for solving complex problems.