Problem 54 - 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. $$\left\\{\begin{aligned} 4 y &=-8 \\ 7 x-2 y &=25 \end{aligned}\right.$$

Step-by-Step Solution

Answer

Without the actual graph, the precise coordinates cannot be provided. However, the process involves using a graphing utility to plot the provided equations, identifying their intersection point, and rounding those coordinates to three decimal places for the final solution.

Step 1: Graph the First Equation

The first equation, \(4y = -8\), can be rewritten in the form of \(y = -2\). When graphing this equation, it's understood that for all possible x-values, y will always be -2. This will result in a horizontal line at \(y = -2\).

Step 2: Graph the Second Equation

The second equation, \(7x - 2y = 25\), can be rewritten in the slope-intercept form \(y = (7/2)x - (25/2)\). This equation represents a line with a slope of 3.5 and y-intercept at -12.5. After graphing this equation on the same coordinate plane, it should intersect the line of the first equation.

Step 3: Find the Intersection Point

The intersection point of the two lines represents the solution for the system. Identify this point on the graph and round the coordinates to three decimal places.

Key Concepts

Linear Equations

Linear equations are an essential concept in mathematics. These are equations that form a straight line when graphed on a coordinate plane. A typical linear equation takes the form of \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. In our exercise, we are dealing with two linear equations:

• The first equation is \(4y = -8\), which simplifies to \(y = -2\). This represents a horizontal line.
• The second equation is \(7x - 2y = 25\).

These linear equations can intersect at a specific point, revealing the solution to the system of equations. Understanding how to manipulate them into different forms, like the slope-intercept form, is key to easily graphing them.

Slope-Intercept Form

The slope-intercept form is a way to express linear equations. It is represented as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This form is particularly useful for graphing because it clearly shows the slope and starting point of the line. In our exercise, the second equation \(7x - 2y = 25\) can be rearranged into \(y = \frac{7}{2}x - \frac{25}{2}\). This expresses the equation in slope-intercept form, revealing the slope \(\frac{7}{2}\) and y-intercept \(-12.5\). Knowing these values allows us to accurately draw the line and understand how it behaves on the graph.

Intersection Point

The intersection point of two lines in a system of equations is crucial because it represents the solution to the system. In simpler terms, it's the point where both equations are true simultaneously, meaning both lines cross at this exact location. After graphing the two lines from our exercise:

• The horizontal line \(y = -2\).
• The other line, \(y = \frac{7}{2}x - \frac{25}{2}\).

The solution is where these two lines meet. Identifying and approximating this point on a graph gives us the answer we're seeking. In the real world and in mathematics, finding this intersection is invaluable as it often represents harmonized results of two different scenarios or systems.

Graphing Utility

A graphing utility is a valuable tool for visualizing mathematical equations, particularly when finding solutions for systems of equations. It allows users to plot graphs accurately, adjust viewing windows, and visually identify important information like intersection points. Using a graphing utility for the equations from our exercise:

• The horizontal line \(y = -2\) is plotted easily.
• The line \(y = \frac{7}{2}x - \frac{25}{2}\) is also graphically represented.

By utilizing this tool, we can find the intersection point of the two lines quickly. Graphing utilities enhance understanding and provide a clear representation of complex systems, taking a lot of the guesswork out of manually sketching the graphs.