Problem 57 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the two equations. Use the graphs to approximate the solution of the system. Round your results to three decimal places. \(\left\\{\begin{array}{l}\frac{1}{3} x+y=-\frac{1}{3} \\ 5 x-3 y=7\end{array}\right.\)

Step-by-Step Solution

Answer

The solution of the system is approximately at the point (x, y) which will be the intersection of these two lines. Remember, this point must be rounded to three decimal places.

Step 1: Graph the first equation

Write the first equation in the form \(y = mx + c\). Thus, the rewritten equation is \(y = -\frac{1}{3}x -\frac{1}{3}\). Plot this equation using a graphing utility.

Step 2: Graph the second equation

Similarly rewrite the second equation as \(y= \frac{5}{3}x - \frac{7}{3}\) and plot this equation on the same graph.

Step 3: Determine the point of intersection

Identify the point where the two lines intersect. That's the solution to this system of equations.

Step 4: Round the solution

The coordinates of the intersection point are the solution of the system. However, as per the original problem, you need to round the solution to three decimal places.

Key Concepts

Graphing Linear Equations

Graphing linear equations is a fundamental technique in algebra and coordinate geometry. It involves plotting a line based on an equation typically written in the form \(y = mx + c\), where \(m\) represents the slope and \(c\) is the y-intercept. This form makes it easier to visualize how changes in the equation affect the graph.

To graph a linear equation, you follow these steps:

• Convert the equation to slope-intercept form, if necessary. This makes plotting simpler.
• Identify the y-intercept \(c\). This is the point where the line crosses the y-axis. Plot this point first.
• Use the slope \(m\), which is the ratio of the rise to the run, to find another point on the line. For example, a slope of \(1/2\) means you move up one unit for every two units you move to the right.
• Draw a straight line through these points, extending it across the graph.

With these steps, you can visualize or use a graphing utility to plot the line accurately. The graphical method helps not only in solving systems of equations but also in understanding the relationship between variables.

Intersection Point

The intersection point in the context of systems of equations is where two lines cross on a graph. This point represents the solution to the system, as it provides the values of \(x\) and \(y\) that satisfy both equations.

To find the intersection point:

• Graph the first equation using the steps outlined for graphing linear equations.
• Graph the second equation on the same coordinate plane.
• Identify the point where the two lines meet. This is done visually or by using a graphing utility.

The coordinates of this intersection point are the solution to the system of equations, meaning both lines share these \(x\) and \(y\) values. If the lines intersect at a distinct point, there's one unique solution. If they're parallel, there is no solution, and if they overlap, there are infinitely many solutions. In practical terms, finding this point allows you to understand how the two equations relate to each other analytically.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometric shapes using a coordinate system. In this branch of mathematics, the position of points is determined based on their coordinates, allowing the understanding of geometric properties using algebra.

A few fundamental concepts include:

• Coordinates: Ordered pairs \((x, y)\) that define the position of points on the Cartesian plane.
• Distance Formula: Used to calculate the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) given by \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
• Midpoint Formula: The midpoint of a segment connecting two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\).
• Slopes of Lines: The slope \(m\) of a line through points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2-y_1}{x_2-x_1}\).

Coordinate geometry helps in understanding the relationship between algebraic concepts and geometric figures. It's particularly useful in solving systems of equations, finding intersection points, and studying various geometrical properties in a logical and ordered manner.