Problem 50 - Precalculus Mathematics 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 rounded to two decimal places, either by zooming in and using [RACE ] 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, rounded to two decimal places, is \((1.41, 0.95)\).

Step 1: Solve for y in the first equation

To graph the equations, we first express \(y\) in terms of \(x\). Start with the first equation: \(18.72x - 14.91y = 12.33\). Solve for \(y\) by rearranging the equation: \[14.91y = 18.72x - 12.33\]Divide through by \(14.91\) to isolate \(y\):\[y = \frac{18.72}{14.91}x - \frac{12.33}{14.91}\]Simplify the fractions to get the equation in slope-intercept form \(y = mx + c\):\[y \approx 1.256x - 0.827\]

Step 2: Solve for y in the second equation

Now, solve the second equation in the same way: \(6.21x - 12.92y = 17.82\). Rearrange to solve for \(y\):\[12.92y = 6.21x - 17.82\]Divide through by \(12.92\) to get:\[y = \frac{6.21}{12.92}x - \frac{17.82}{12.92}\]Simplify:\[y \approx 0.481x - 1.38\]

Step 3: Graph the equations

Use a graphing calculator or software. Enter the two equations:Equation 1: \(y = 1.256x - 0.827\)Equation 2: \(y = 0.481x - 1.38\)Ensure both equations appear in the same viewing window for comparison. Adjust the window settings if necessary to see where the lines intersect.

Step 4: Find the intersection point

Use the 'Intersect' feature on your graphing device. Select both lines and follow the prompts to calculate the intersection point. When prompted, ensure you zoom in adequately to see the precise intersection. The device will provide the coordinates of the intersection, which you should round to two decimal places.

Step 5: Check the result

Verify that the intersection point calculated actually satisfies both original equations by substituting the x and y values back into each. Small errors may occur in the graphing process, so this ensures accuracy. If the calculated point fits both equations, it is the solution to the system.

Key Concepts

Solving for y

A common requirement in solving systems of linear equations is to express each equation in terms of the variable, typically "y". This process is known as "solving for y". When we have an equation like \(18.72x - 14.91y = 12.33\), our goal is to isolate "y" on one side. Here's how you do it:

• First, rearrange the equation so that all terms involving "y" are on one side: \(14.91y = 18.72x - 12.33\).
• Then, divide every term by the coefficient of "y" (in this case, 14.91) to solve for "y": \(y = \frac{18.72}{14.91}x - \frac{12.33}{14.91}\).
• Finally, simplify the fractions to arrive at a cleaner expression.

This process is repeated for each equation in the system. Once all equations are expressed in terms of "y", they can be easily graphed.

Intersection of Lines

The intersection of lines is a key concept when solving systems of linear equations graphically. When two lines intersect, the point where they meet represents the common solution to both equations. To find the intersection:

• Graph each equation on the same set of axes.
• Visually or using a tool, determine where the lines cross.
• This point of crossing is the solution to the system, expressed as a pair of coordinates \((x, y)\).

For precision, using a graphing calculator's "Intersect" function can help pinpoint this location accurately and is particularly useful when the lines seem close but don't clearly cross at integer values. This calculation is often an approximation, rounded to two decimal places, to ensure clarity.

Slope-Intercept Form

The slope-intercept form of a linear equation is essential for graphing and understanding the behavior of lines. It is generally expressed as \(y = mx + b\), where \(m\) denotes the slope and \(b\) is the y-intercept. The advantages of this form include:

• It directly shows the slope, \(m\), which indicates the steepness and direction of the line.
• The y-intercept, \(b\), reveals where the line crosses the y-axis, allowing for quick graphing from the intercept point.

For instance, a line given by \(y = 1.256x - 0.827\) has a slope of 1.256, which means it rises 1.256 units vertically for every unit it moves horizontally. This form is vital for quickly plotting and understanding the underlying relationship depicted by the linear equation.

Graphing Calculator

Graphing calculators are powerful tools for solving systems of equations visually. They allow for quick plotting and analysis of intersecting lines. When using a graphing calculator:

• First, ensure your equations are in slope-intercept form: \(y = mx + b\). This preparation simplifies the process of entering equations into the device.
• Enter each equation carefully, and adjust the viewing window or range to ensure visibility of the intersection point.
• Use features such as "Zoom" to refine the view or "Intersect" to accurately find the meeting point of the lines.

Graphing calculators facilitate an interactive approach to verifying solutions. They also help visualize mathematical concepts, providing a deeper understanding beyond numerical calculations.