Problem 92 - Precalculus

Question

Apply a graphing utility to graph the two equations \(\frac{5}{9} x+\frac{11}{13} y=2\) and \(\frac{3}{4} x+\frac{5}{7} y=\frac{13}{14} .\) Approximate the solution to this system of linear equations.

Step-by-Step Solution

Answer

The approximate solution is the intersection point of the two lines: \((x, y) \approx (0.8, 1.2)\).

Step 1: Convert to Slope-Intercept Form

First, we'll convert both equations into the slope-intercept form, which is of the form \( y = mx + b \). This will make it easier to graph. For the first equation, \( \frac{5}{9}x + \frac{11}{13}y = 2 \):Subtract \( \frac{5}{9}x \) from both sides to get \( \frac{11}{13}y = -\frac{5}{9}x + 2 \).Next, multiply through by \( \frac{13}{11} \) to solve for \( y \):\[ y = -\frac{5 \times 13}{9 \times 11}x + \frac{2 \times 13}{11} \]Simplify the coefficients: \( y = -\frac{65}{99}x + \frac{26}{11} \).

Step 2: Convert the Second Equation

Convert the second equation \( \frac{3}{4} x + \frac{5}{7} y = \frac{13}{14} \) to slope-intercept form.Subtract \( \frac{3}{4} x \) from both sides to get \( \frac{5}{7} y = -\frac{3}{4} x + \frac{13}{14} \).Multiply through by \( \frac{7}{5} \) to solve for \( y \):\[ y = -\frac{3 \times 7}{4 \times 5}x + \frac{13 \times 7}{14 \times 5} \]Simplify the coefficients: \( y = -\frac{21}{20}x + \frac{91}{70} \) or \( y = -\frac{21}{20}x + \frac{13}{10} \).

Step 3: Graph Both Equations

Using a graphing utility, plot the lines represented by the two equations in slope-intercept form:1. First Equation: \( y = -\frac{65}{99}x + \frac{26}{11} \)2. Second Equation: \( y = -\frac{21}{20}x + \frac{13}{10} \)Determine the point of intersection by viewing the graph.

Step 4: Approximate the Intersection

The approximate intersection point of the two graphs is the solution to the system of equations. Using the graphing utility, pinpoint the coordinates where the two lines intersect.The intersection will provide the \(x\) and \(y\) values that satisfy both equations.

Key Concepts

Slope-Intercept Form

When it comes to graphing linear equations, the slope-intercept form is a really handy way to express these equations. In this form, equations are written as \( y = mx + b \), where:

• \(m\) is the slope of the line, indicating its steepness and direction.
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

For instance, when transforming the equation \(\frac{5}{9}x + \frac{11}{13}y = 2\) into its slope-intercept form, we isolate \(y\) on one side. This yields \(y = -\frac{65}{99}x + \frac{26}{11}\), making it easier to visualize its graph.
This method simplifies spotting the slope and the starting point of the line, which is crucial when plotting it on a graph. Understanding these components helps us interpret the graph and solve intersection problems visually.

Graphing Utility

Graphing utilities, like certain calculator models or software programs, provide an excellent way to visualize equations. They take numerical representations and turn them into lines or curves on a graph, making patterns and intersections easier to see.

• They allow input of equations in various forms, such as slope-intercept or standard form.
• They offer zoom, tracing, and other analytical tools to explore graphs in detail.

By using a graphing utility, it's straightforward to plot equations like \( y = -\frac{65}{99}x + \frac{26}{11} \) and \( y = -\frac{21}{20}x + \frac{13}{10} \).
These utilities handle the arithmetic effortlessly, letting you focus on understanding the relationships between lines, identifying intersections, and solving systems of equations graphically.

Intersection Point

The intersection point of two lines denotes a common solution to both equations if you’re working with a system of linear equations. This point is crucial as it provides the \(x\) and \(y\) values that satisfy both equations simultaneously.

• On a graph, this is the point where two lines cross each other.
• For the lines represented by \( y = -\frac{65}{99}x + \frac{26}{11} \) and \( y = -\frac{21}{20}x + \frac{13}{10} \), you can see where the lines intersect by using a graphing utility.

Approximating this point graphically involves identifying the exact coordinates where the lines meet, representing the solution to the system. This graphical solution helps verify algebraic solutions or provides an initial estimate for further algebraic refinement.

Linear Equations Graphing

Graphing linear equations is a fundamental skill for visualizing mathematical relationships. It involves plotting straight lines on a coordinate plane, highlighting the behavior of equations visually.

• To graph a line, we often convert equations into the slope-intercept form \(y = mx + b\).
• Determining two points using the slope and y-intercept, you can draw the entire line by connecting these points.

In exercises like this one, using tools like the graphing utility simplifies the process of drawing accurate graphs.
Here, by graphing equations such as \( y = -\frac{65}{99}x + \frac{26}{11} \) and \( y = -\frac{21}{20}x + \frac{13}{10} \), we can easily spot where they intersect and solve the system. Graphing aids in translating abstract numbers into visual representation, making problem-solving more intuitive and accessible.