Problem 38 - Precalculus: Mathematical for Calculus

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. $$\left\\{\begin{aligned} -435 x+912 y &=0 \\ 132 x+455 y &=994 \end{aligned}\right.$$

Step-by-Step Solution

Answer

(2.17, 1.03) to two decimal places.

Step 1: Solve for y in both equations

For the first equation, \(-435x + 912y = 0\), solve for \(y\):\[y = \frac{435}{912}x = 0.4763x.\]For the second equation, \(132x + 455y = 994\), solve for \(y\): \[y = \frac{994 - 132x}{455} = 2.1846 - 0.2890x.\]

Step 2: Graph both equations

Using a graphing calculator or software, input the equations obtained in Step 1:\[y_1 = 0.4763x\]\[y_2 = 2.1846 - 0.2890x\]Graph both equations in the same viewing rectangle.

Step 3: Locate the Intersection Point

Use the graphing calculator's `Intersect` function or `TRACE` method to find the coordinates of the intersection point of the two lines. This will be the solution to the system.

Step 4: Verify the Intersection Coordinates

Ensure that the coordinates obtained are correct to two decimal places, noting any values displayed on the graphing calculator or software.

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations that are solved together to find a common solution. These equations can represent real-world problems such as finding the point where two different paths cross. In this context, each equation describes a line in a coordinate plane.
In solving a system of linear equations, you are essentially looking for a point

• where all the equations in the system intersect or meet.
• This common point, if it exists, is referred to as the solution of the system.

Multiple methods exist for solving these systems, such as substitution, elimination, and graphing. In graphing, each equation is plotted on a coordinate plane to visually identify the intersecting point. This approach is instructive as it provides a graphical representation of the solution.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic equations to describe geometric figures and spatial relationships. In the context of linear equations, coordinate geometry helps visualize the behavior of lines through graphs. These graphs are plotted on a Cartesian coordinate system.
A Cartesian coordinate system consists of a horizontal axis (x-axis) and a vertical axis (y-axis) intersecting at a point called the origin. Linear equations can be expressed in the slope-intercept form:

• For example, an equation like \(y = mx + c\) includes the slope \(m\), representing how steep the line is, and \(c\), the y-intercept, indicating where the line crosses the y-axis.
• Plotting this line involves choosing values for \(x\), calculating the corresponding \(y\) values, and marking these points on the graph.

By understanding coordinate geometry, students can easily shift between algebraic expressions and graphical representations, solidifying their grasp of the system of equations and the concept of solutions as intersection points.

Graphing Calculator Usage

A graphing calculator is a useful tool in visualizing and solving systems of linear equations. These calculators have the capability to plot graphs based on inputted algebraic equations, removing the guesswork involved in manual graphing.
Here's how you can optimize graphing calculator usage for systems of linear equations:

• **Input Equations**: Always arrange each equation in terms of \(y\) before inputting into the calculator. This is key for visualization.
• **Graph the System**: Use the calculator to graph both lines in the same viewing space. This helps in easily identifying points of intersection.
• **Finding the Intersection**: Utilize the calculator's 'Intersect' function or 'TRACE' feature. Zoom in to get more accurate readings of the coordinates where the lines meet, ensuring precision up to two decimal places as required.
• **Verification**: It's a good practice to verify the calculated intersection to ensure the solution's accuracy and reliability.

Adopting these methods enhances comprehension and allows students to solve linear system equations more efficiently and accurately.