Problem 35 - 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. $$\left\\{\begin{array}{l} 0.21 x+3.17 y=9.51 \\ 2.35 x-1.17 y=5.89 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system is approximately \((2.06, 2.35)\).

Step 1: Solve for y in terms of x

For the equation \(0.21x + 3.17y = 9.51\), solve for \(y\) by subtracting \(0.21x\) from both sides, resulting in \(3.17y = 9.51 - 0.21x\). Then, divide everything by 3.17 to isolate \(y\), which gives \(y = \frac{9.51 - 0.21x}{3.17}\). For the equation \(2.35x - 1.17y = 5.89\), solve for \(y\) by subtracting \(2.35x\) from both sides, resulting in \(-1.17y = 5.89 - 2.35x\). Then, divide everything by \(-1.17\) to isolate \(y\), which gives \(y = \frac{2.35x - 5.89}{1.17}\).

Step 2: Graph the Lines

Use your graphing device to plot both equations. Enter the equations for \(y\) from Step 1 into your graphing calculator or software. Ensure they are graphed over the same viewing rectangle for accurate comparison.

Step 3: Locate Intersection Point

Use the graphing device's intersect feature or zoom and trace functions to pinpoint where the two lines cross. This intersection represents the solution to the system of equations.

Step 4: Determine Exact Intersection Coordinates

After locating the intersection, use the graphing calculator's INTERSECT function to precisely find the coordinates. The x and y values are the solution to the system, rounded to two decimal places.

Key Concepts

Solving Systems of Equations

Solving systems of equations involves finding the values of variables that satisfy all equations simultaneously. When you have two linear equations, they can be represented as lines on a graph. To find the solution, or intersection point, both lines need to be solved for one variable, typically y, in terms of the other variable, x. Once in this form, you can graph each equation to visualize where they meet. These meeting points are crucial as they represent the exact values where both conditions of the system of equations are fulfilled. For our problem, you'll first transform each equation into the y = mx + b format: - Equation 1 becomes: \[y = \frac{9.51 - 0.21x}{3.17}\]- Equation 2 becomes: \[y = \frac{2.35x - 5.89}{1.17}\]These transformations allow us to graph each line and give a clear picture of where they intersect, solving the system visually.

Intersection of Lines

The intersection of lines on a graph is a fundamental concept in finding solutions to systems of linear equations. When two lines intersect, they share a common point. This shared point is essential because it provides the values of x and y that satisfy both equations. Graphically, the intersection is the point at which the two lines cross. For our given equations, you would plot both lines on the same axes using their respective y-form equations. Important facts about line intersections:

• If the lines intersect, the system has a single solution, represented by the precise intersection point.
• If the lines are parallel, they do not intersect, implying zero solutions unless they overlap completely, indicating infinite solutions.
• The intersection point provides the exact values needed to solve the system, found graphically as coordinates such as (x, y).

Graphing Calculator Techniques

Using a graphing calculator can greatly simplify the process of solving systems of equations. Once you have equations in y = mx + b form, you can input them directly into the calculator. Here are some tips for effective use:

• Input each equation separately, ensuring no errors in coefficients or signs.
• Select an appropriate viewing window so that both lines are visible and any intersections are clearly shown.
• Use the [INTERSECT] feature on the calculator. This tool helps you accurately find the intersection point you're interested in without manual estimation.
• If your calculator does not have an intercept feature, use the [TRACE] mode. This allows you to move along the graph and closely examine points where lines converge.
• Finally, zoom in on the intersection to ensure precision, and once identified, extract the x and y coordinates to your specified decimal place.

By mastering these techniques, you can efficiently solve even complex systems and understand the corresponding graphical representations.