Problem 79 - Introductory Algebra for College Students

Question

Read Exercise \(72 .\) Then use a graphing utility to solve the systems. $$\left\\{\begin{array}{l}y=\frac{1}{3} x+\frac{2}{3} \\ y=\frac{5}{7} x-2\end{array}\right.$$

Step-by-Step Solution

Answer

The exact point of intersection (solution of the system of linear equations) can be identified using a graphing utility. The task necessitates the use of this tool to provide an accurate solution.

Step 1: Graph the first equation

First, graph the line defined by the equation \(y = \frac{1}{3}x + \frac{2}{3}\). This line has a slope of \(\frac{1}{3}\) and y-intercept of \(\frac{2}{3}\). Starting at the y-intercept, go up 1 unit and to the right 3 units to draw the line.

Step 2: Graph the second equation

Next, graph the line defined by the equation \(y = \frac{5}{7}x - 2\). This line has a slope of \(\frac{5}{7}\) and a y-intercept of -2. Starting at the y-intercept, go up 5 units and to the right 7 units to draw the line.

Step 3: Identify intersection point

The graphs of the two lines will cross at a certain point. This point represents the solution to the system of equations. Use the graphing utility to recognize precisely where this point of intersection is.

Key Concepts

Graphing Equations

Graphing equations is a fundamental skill in algebra and math. It helps visualize relationships between variables, making it easier to understand how they interact. When you graph equations, each point on the graph represents a solution to that equation. In the context of a system of equations, graphing is particularly useful because it can visually show where the equations intersect. This point of intersection is where both equations are true simultaneously, giving us the solution to the system of equations.

• To graph a linear equation, you typically need its slope and y-intercept.
• The graph of the equation will be a straight line on the coordinate plane.
• By plotting at least two points and drawing a line through them, you can represent the entire set of solutions for the equation.

Breaking down graphing further, it's not just about drawing. It's about understanding how changes in the equation affect the graph, and vice versa. This connection is crucial, especially when tackling systems of equations.

Slope-Intercept Form

The slope-intercept form of a linear equation is a way to express the equation of a line. It is written as \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept. This form is beneficial because it quickly reveals key characteristics of the line, allowing for straightforward graphing.

• The slope \( m \) indicates the steepness of the line and the direction it tilts. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls.
• The y-intercept \( b \) is where the line crosses the y-axis. This is the point you start plotting the line on a graph.

For example, in the equation \( y = \frac{1}{3}x + \frac{2}{3} \), the slope is \( \frac{1}{3} \) and the y-intercept is \( \frac{2}{3} \). This means you start plotting at \( (0, \frac{2}{3}) \) and use the slope to find another point by moving up 1 unit and right 3 units. Understanding slope-intercept form simplifies graphing and helps in solving systems of equations.

Graphing Utilities

Graphing utilities are tools that assist with creating graphs of equations. They range from simple hand-drawn graphs to sophisticated software applications that can handle complex systems. These utilities are essential for tasks that require precision, such as identifying intersection points in a system of equations.

• Graphing calculators and software can quickly plot graphs, making them invaluable for checking work or exploring graphs that are difficult to draw by hand.
• They allow you to input equations and instantly see their graphs, saving time and reducing the chance of errors.
• Graphing utilities can also provide numerical solutions to intersection points, which is extremely useful when working with systems of equations that don’t intersect at integer coordinates.

Utilizing graphing utilities can deepen your understanding by allowing you to experiment with different equations. You can see how changes in the equations affect their graphs and solutions. This hands-on interaction empowers you to better grasp the concepts of graphing and solving systems of equations.