Problem 36 - 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. $$\left\\{\begin{aligned} 18.72 x-14.91 y &=12.33 \\ 6.21 x-12.92 y &=17.82 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The intersection point, and therefore the solution, is approximately (0.83, 0.21).

Step 1: Isolate y in the First Equation

To graph the equations, solve for \( y \) in terms of \( x \). Start with the first equation: \( 18.72x - 14.91y = 12.33 \). Move \( 18.72x \) to the right side: \( -14.91y = -18.72x + 12.33 \). Then, divide by \( -14.91 \): \( y = \frac{18.72}{14.91}x - \frac{12.33}{14.91} \). Simplifying, we get \( y = 1.25x - 0.83 \).

Step 2: Isolate y in the Second Equation

For the second equation \( 6.21x - 12.92y = 17.82 \), move \( 6.21x \) to the right side: \( -12.92y = -6.21x + 17.82 \). Divide by \( -12.92 \): \( y = \frac{6.21}{12.92}x - \frac{17.82}{12.92} \). Simplifying, this becomes \( y = 0.48x - 1.38 \).

Step 3: Graph the Equations

Using a graphing device, plot the equations: \( y = 1.25x - 0.83 \) and \( y = 0.48x - 1.38 \) on the same graph. Adjust the viewing window to see both lines clearly.

Step 4: Find the Intersection Point

Use the "Intersect" function on your graphing device to find the point where these two lines cross. This involves selecting both graph lines and then finding their intersection point.

Step 5: Verify and Note the Intersection

Ensure the device is set to provide results to two decimal places. The intersection is where the solution to the system of equations is located. Note down the coordinates of this point.

Key Concepts

System of Equations

A system of equations refers to a set of two or more equations with the same variables. In this task, we are dealing with a system consisting of two linear equations. Both equations connect through the variables, specifically relying on the relationship between them to find a solution. The objective is to find the specific values of these variables, like the value of \( x \) and \( y \), that satisfy both equations simultaneously.

Typically, there are several methods to solve a system of equations, such as substitution, elimination, or graphical methods. In this context, we are using graphing to identify where the lines represented by each equation intersect. When the lines intersect, it gives us the solution to the system, where the same \( x \) and \( y \) values solve both equations. By understanding how these systems form and function, you'll unlock the ability to predict and analyze relationships in all sorts of linear contexts.

Solving for Y

Before graphing lines using a graphing calculator, it's essential to have them in a format that the calculator understands, typically expressed as \( y = mx + b \). This is known as the slope-intercept form. Solving for \( y \) means rearranging the equation so that \( y \) stands alone on one side of the equation.

Here's how we can solve for \( y \):

• Take the first equation \(18.72x - 14.91y = 12.33\). Move \(18.72x\) to the right by subtracting it from both sides.
• Simplify: \(-14.91y = -18.72x + 12.33\).
• Finally, divide every term by \(-14.91\) to isolate \( y \).

After simplifying, this gives us \( y = 1.25x - 0.83 \), which tells us how \( y \) changes with \( x \). Follow a similar process for the second equation to get \( y = 0.48x - 1.38 \). Having both equations in this form makes graphing straightforward on a calculator.

Intersection of Lines

The intersection of lines in a system of equations gives the solution to the system. When lines intersect on a graph, they share a common point, which represents the values of \( x \) and \( y \) that satisfy both equations.

To find the intersection:

• Graph both lines using their slope-intercept forms. This involves plotting points based on the mathematical relationships \( y = 1.25x - 0.83 \) and \( y = 0.48x - 1.38 \).
• Look for the point where the lines cross each other on the graph. This crossing is the intersection and hence the solution.
• Using the graphing calculator's intersect function can help pinpoint this exact spot more accurately, often shown as coordinates.

This graphical method of finding intersections illustrates visually how equations relate to each other, emphasizing the significance of solutions as connection points between seemingly distinct linear expressions.

Using Graphing Calculators

Graphing calculators are powerful tools for visualizing equations, making the solving process more intuitive and precise. These devices allow for easy manipulation and understanding of graphs, enhancing the learning process by enabling real-time visualization of algebraic expressions.

To use a graphing calculator effectively:

• First, input each equation in its solved form for \( y \).
• Position the viewing window so you can see where both lines are likely to intersect. Adjusting the window helps in visualizing the solution better.
• Employ tools like "Trace" or "Intersect" to narrowly focus on the crossing point of the lines, giving accurate intersection points up to the required decimal places.

Graphing calculators streamline solving systems of linear equations, especially useful for complex calculations or when an exact visual match is necessary. This technological aid simplifies understanding and interpreting results efficiently.