Problem 49 - 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. $$\left\\{\begin{array}{l}0.21 x+3.17 y=9.51 \\\2.35 x-1.17 y=5.89\end{array}\right.$$

Step-by-Step Solution

Answer

The intersecting point, rounded to two decimal places, is the solution: \( (2.65, 2.82) \).

Step 1: Solve the First Equation for y

To solve the equation \(0.21x + 3.17y = 9.51\) for \(y\), first isolate \(y\) by subtracting \(0.21x\) from both sides, obtaining \(3.17y = -0.21x + 9.51\). Then divide every term by 3.17 to get \(y = -\frac{0.21}{3.17}x + \frac{9.51}{3.17}\). This simplifies to \(y = -0.066x + 3.00\).

Step 2: Solve the Second Equation for y

Solve the equation \(2.35x - 1.17y = 5.89\) for \(y\). Subtract \(2.35x\) from both sides to get \(-1.17y = -2.35x + 5.89\). Next, divide every term by -1.17, resulting in \(y = \frac{2.35}{1.17}x - \frac{5.89}{1.17}\). This simplifies to \(y = 2.01x - 5.03\).

Step 3: Graph Both Equations

Using a graphing device, graph the two equations determined in Steps 1 and 2: \(y = -0.066x + 3.00\) and \(y = 2.01x - 5.03\). Observe the point where the two lines intersect.

Step 4: Use Intersect Feature to Find Solution

Utilize the "Intersect" feature on your graphing device. This will calculate the point where the two lines, \(y = -0.066x + 3.00\) and \(y = 2.01x - 5.03\), cross. Ensure the results are rounded to two decimal places.

Step 5: Verify and Interpret the Solution

The intersection point obtained using the "Intersect" feature will give the values of \(x\) and \(y\) that solve the system of equations. Verify that substituting these values back into the original equations satisfies both equations.

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool that helps us visualize mathematical concepts, like systems of linear equations. It can display graphs of equations, making it easier to see where they intersect.

To begin using a graphing calculator, you need to input equations in a format it understands, often in the form of \( y = mx + b \). This involves solving for \( y \) first, which we'll touch on in detail later. After inputting the equations, you can view them as lines on the graph.

• Use the zoom function to get a closer look at specific areas of your graph.
• The "Intersect" feature allows you to find the exact point where two lines cross.

Combining these tools makes a graphing calculator essential for quickly finding solutions.

Intersection of Lines

The intersection of lines is an important concept when dealing with systems of linear equations. This is the point where two lines cross each other on a graph, indicating that the two equations share common values for \( x \) and \( y \).

When you graph two equations using their \( y = mx + b \) form, the intersection point gives you the solution to the system. At this point, both equations are satisfied with the same \( x \) and \( y \) values.

• The solution is the set of coordinates \( (x, y) \) where both lines meet.
• If the lines are parallel, they will not intersect, indicating no solution.
• If the lines are the same, they have infinite intersection points, meaning infinite solutions exist.

Finding this intersection manually can be cumbersome, but a graphing calculator's "Intersect" feature simplifies the process.

Solving for y

When working with systems of linear equations on a graphing calculator, it's crucial to solve each equation for \( y \) to put them in slope-intercept form, \( y = mx + b \). This is necessary because the graphing calculator needs this format to graph the equations properly.

Steps to solve for \( y \) include rearranging the equation and simplifying it to isolate \( y \). For example, if you have the equation \( 0.21x + 3.17y = 9.51 \):

• Subtract \( 0.21x \) from both sides to start isolating \( y \).
• Divide all terms by the coefficient of \( y \), which is \( 3.17 \), to completely isolate \( y \).
• Repeat this process for any other equation you have.

Once in the correct format, you can input your equations into a graphing calculator to visualize the solution.