Problem 51 - Algebra and Trigonometry

Question

\(49-52\) Use a graphing device to graph both lines in the same viewing rectangle. (Note that you must solve for \(y\) in terms of \(x\) before graphing if you are using a graphing calculator.) Solve the system rounded to two decimal places, either by zooming in and using [TRACE] or by using Intersect. $$ \left\\{\begin{array}{l}{2371 x-6552 y=13,591} \\ {9815 x+992 y=618,555}\end{array}\right. $$

Step-by-Step Solution

Answer

Find the intersection point of the lines using a graphing device, rounded to two decimal places.

Step 1: Rewrite the Equations in Slope-Intercept Form

The slope-intercept form of a line is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Start by solving each equation for \( y \). 1. For the first equation, \( 2371x - 6552y = 13591 \): \[ 6552y = 2371x - 13591 \] Divide every term by 6552: \[ y = \frac{2371}{6552}x - \frac{13591}{6552} \]2. For the second equation, \( 9815x + 992y = 618555 \): \[ 992y = -9815x + 618555 \] Divide every term by 992: \[ y = -\frac{9815}{992}x + \frac{618555}{992} \]

Step 2: Graph the Lines

Using the equations from Step 1, graph both lines on the same coordinate plane using a graphing device (such as a graphing calculator or software). The equations are:\[ y = \frac{2371}{6552}x - \frac{13591}{6552} \]\[ y = -\frac{9815}{992}x + \frac{618555}{992} \]Ensure the viewing window includes the intersection point of these two lines.

Step 3: Find the Intersection Point

Use the 'Intersect' feature or 'TRACE' function on your graphing device to determine where the two lines intersect. This intersection point is the solution to the system of equations. Ensure to round the x and y coordinates to two decimal places.

Key Concepts

Graphing Lines

Graphing lines is a fundamental concept in solving systems of linear equations. When you graph a line, you are plotting all the points that satisfy a linear equation in two variables such as x and y. This is typically depicted as a straight line on a Cartesian plane. To start graphing, you need the equation of the line in a form that's easy to use on graphing devices.

• One common approach is to rearrange the equation into what's called slope-intercept form.
• This involves solving for y to get the equation into the form y = mx + b.

Once you have the equation in slope-intercept form, you can plot the y-intercept (b), which is where the line crosses the y-axis. From there, using the slope (m), you determine the rise over run — moving up or down and side to side to find other points on the line.

Graphing both lines from a system of equations together is crucial because it allows you to visually identify where the lines intersect, which leads to finding the solution for both equations.

Slope-Intercept Form

The slope-intercept form is one of the most useful forms for the equation of a line. It is expressed as y = mx + b, where m represents the slope and b represents the y-intercept.

• The slope, m, shows the steepness and direction of a line. A positive slope means the line rises, while a negative slope means it falls.
• The y-intercept, b, denotes the point where the line crosses the y-axis. This is of the form (0, b) in coordinates.

This form is advantageous because it gives clear information about the graph's characteristics, allowing you to easily draft the graph by plotting the y-intercept and then applying the slope to find additional points.

In the context of systems of equations, rewriting each equation in slope-intercept form is an essential first step. It prepares the equations to be graphed and analyzed for potential solutions. The original problem involves converting complex equations to this form, which simplifies the graphing process.

Intersection Point

Finding the intersection point is the key to solving a system of linear equations graphically. This point is where both lines cross on a graph, indicating a single pair of (x, y) values that make both equations true simultaneously.

• To find this intersection, you can use a graphing tool's 'Intersect' feature or manually use the 'TRACE' functionality to pinpoint the coordinates.
• Once found, it's important to note these coordinates as they represent the solution to the system.

In practical applications, knowing how to identify the intersection point on a graph helps bridge an understanding from graphical analysis to algebraic calculations. The intersection not only showcases the graphical solution but also relates back to checking your work mathematically, ensuring coherence between the graph and equations.

In the context of the exercise, rounding the coordinates of the intersection to two decimal places ensures a precise answer is obtained, while within an acceptable margin of error, typical in many real-world scenarios.