Problem 35 - Algebra and Trigonometry

Question

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 correct to two decimal places, either by zooming in and using TRACE or by using Intersect. $$\left\\{\begin{array}{l}{0.21 x+3.17 y=9.51} \\ {2.35 x-1.17 y=5.89}\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system is \( x \approx 2.00 \), \( y \approx 2.87 \).

Step 1: Solve for y in the First Equation

To graph the equations, we need them in the form \( y = mx + c \). Start with \( 0.21x + 3.17y = 9.51 \). Solve for \( y \):\[ 3.17y = -0.21x + 9.51 \]\[ y = \frac{-0.21}{3.17}x + \frac{9.51}{3.17} \]After computing these values:\[ y = -0.0663x + 3.00 \]

Step 2: Solve for y in the Second Equation

Now take the second equation \( 2.35x - 1.17y = 5.89 \) and solve for \( y \):\[ -1.17y = -2.35x + 5.89 \]\[ y = \frac{2.35}{1.17}x - \frac{5.89}{1.17} \]After computing these values:\[ y = 2.009x - 5.03 \]

Step 3: Graph the Equations

Using a graphing calculator or software, input the two equations derived:1. \( y = -0.0663x + 3.00 \)2. \( y = 2.009x - 5.03 \)Ensure both equations are visible in the same viewing rectangle.

Step 4: Find the Intersection Point

Use the "Intersect" or "Trace" feature of the graphing tool to find where the two lines intersect. Zoom in if necessary to get a more accurate reading. The intersection point will provide the values of \( x \) and \( y \) that satisfy both equations simultaneously.

Step 5: Record the Solution

Once the intersection point is found, record the coordinates \((x, y)\) to two decimal places. These coordinates are the solution to the system of equations.

Key Concepts

Graphing Linear Equations

Graphing linear equations is a fundamental concept in algebra that involves plotting a straight line based on an equation. Each equation represents a linear relationship between two variables, typically denoted as \(x\) and \(y\). In order to graph a linear equation, you need to rearrange it to the slope-intercept form, which is \(y = mx + c\). Here, \(m\) is the slope of the line, representing its steepness or incline, and \(c\) is the y-intercept, indicating where the line crosses the y-axis.

When graphing, you calculate at least two points by choosing values of \(x\) and finding the corresponding \(y\) values using the equation. These points are plotted on a coordinate grid, and a straight line is drawn through them. With the equations provided, you first solve for \(y\) in terms of \(x\) to express them in the appropriate form, making them easier to input into a graphing tool or sketch by hand.

This structured approach helps visualize the relationship between variables and is the basis for solving linear systems by graphing.

Solving Systems by Graphing

Solving systems by graphing involves determining the point at which two or more lines on a graph intersect. This method provides a visual solution to systems of equations. To start, each equation in the system needs to be transformed into the slope-intercept form \(y = mx + c\). Once in this form, you can plot each line on the same coordinate plane.

The intersection of these lines represents the solution to the system of equations. This is because the intersection point is where both equations are satisfied simultaneously. For example, a solution \((x, y)\) means that substituting \(x\) and \(y\) into both original equations will hold true.

Solving systems by graphing provides an intuitive understanding of how different linear relationships interact. However, it is critical to ensure accuracy when graphing to correctly identify the intersection point, especially if decimal precision is required.

Intersection of Lines

The intersection of lines is the cornerstone of solving linear systems graphically. When two lines intersect on a graph, their crossing point is unique and represents the values of \(x\) and \(y\) that satisfy both equations simultaneously. In our example, once the equations are graphed, you will look for the crossing point of the two lines.

Finding the intersection point can be approached manually by drawing accurate graphs or more precisely through technology. By using graphing tools, you can employ functionalities like "Trace" or "Intersect" to zoom in and pinpoint the exact values. The intersection generally results in a pair of coordinates \((x, y)\), which is recorded to the desired precision, often to two decimal places in exercises like this.

Understanding the intersection conceptively solidifies your comprehension of linear relationships, highlighting where two or more conditions meet or coexist.

Graphing Calculator Usage

A graphing calculator is a powerful tool for visualizing mathematical equations and solving problems involving systems of equations. These devices allow you to input equations easily after converting them into the slope-intercept form \(y = mx + c\). Once entered, the calculator graphs the lines, providing a visual representation.

Graphing calculators typically feature functions such as "Intersect" and "Trace." These are invaluable for identifying the intersection point of graphed lines. Users can navigate across the graph to follow the lines and cross-check their meeting point. With the "Zoom" function, you can enhance visibility and gain more precise readings, which is especially helpful when working with decimal solutions.

By mastering your graphing calculator, you can efficiently and accurately solve systems of linear equations, reinforcing your understanding and verification of math problems.