Problem 119 - College Algebra Essentials

Question

Use your graphing utility to enter each side of the equation separately under \(y_{1}\) and \(y_{2} .\) Then use the utility's TABLE or GRAPH feature to solve the equation. \(\frac{2 x-1}{3}-\frac{x-5}{6}=\frac{x-3}{4}\)

Step-by-Step Solution

Answer

In this problem, the solution is found by graphing the two sides of the original equation. Using a graphing utility, you graph these formulas and find the point of intersection. The x-coordinate of this point is the solution to the original equation.

Step 1: Rearrange Equation into Two Formulas

Firstly, we'll rearrange the original equation \(\frac{2 x-1}{3}-\frac{x-5}{6}=\frac{x-3}{4}\) into two equations: a. \(y_{1}= \frac{2 x-1}{3}-\frac{x-5}{6}\) and b. \(y_{2} = \frac{x-3}{4}\). We intend to graph these two expressions under \(y_{1}\) and \(y_{2}\) respectively. On the graph, the solution to the equation is where the two graphs intersect.

Step 2: Input Equations into Graphing Utility

Next, enter the formulas found in step 1 into the graphing utility. Ensure to label them as \(y_{1}\) and \(y_{2}\) as appropriate.

Step 3: Run the GRAPH or TABLE Feature

Run the GRAPH or TABLE feature on your graphing utility, this will produce the graphs (or tables) of \(y_{1}\) and \(y_{2}\).

Step 4: Locate Intersection

The final step is to find where the two graphs intersect. The x-coordinate of this intersection point is the solution to the original equation. Depending on your graphing utility, you might be able to use a 'Trace' or 'Intersection' function to pinpoint this intersection.

Key Concepts

Graphing Utility

Graphing utilities are incredible tools that can make solving equations much easier. They allow you to visualize equations graphically by plotting them on a coordinate plane. This visual representation helps in understanding complex equations that might otherwise seem daunting. A graphing utility can be a calculator or a computer software designed to display graphs.

To use a graphing utility:

• Input each equation separately.
• Choose to display them on the graph.
• Look for tools within the utility to assist finding solutions, like table views or tracing functions.

Graphing utilities make it simple to see where lines or curves intersect, and they often offer features for zooming in on these intersection points for more precise readings.

Intersection Point

In the context of graphing equations, an intersection point is where two graphs meet or cross each other on a coordinate plane. It is a vital concept when solving systems of equations graphically.

Understanding the intersection point:

• The coordinate, often in the form of \(x, y\), shows the solution to the system of equations.
• For the given problem, the x-coordinate of this intersection is the precise value solving the equation.
• Depending on your graphing utility, it may offer an 'Intersection' function to automatically find this point without manual searching.

Thus, locating the intersection point graphically is effective when algebraic methods become cumbersome.

Rational Equations

Rational equations involve fractions where the numerator and/or denominator contain algebraic expressions or variables. They can sometimes be tricky because these equations require managing fractions and finding a common denominator if solved algebraically.

When dealing with rational equations:

• Look for terms that can simplify the fractions before solving.
• Recognize that graphing these can relieve the burden of algebraic manipulation, providing a visual solution.
• Understanding how to manipulate fractions helps prepare these equations for easy graph input.

Graphically solving rational equations, as in \( \frac{2x-1}{3} - \frac{x-5}{6} = \frac{x-3}{4} \), allows one to see their solutions more intuitively, bypassing potential fraction management errors.