Problem 52 - 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. $$\left\\{\begin{aligned}-435 x+912 y &=0 \\\132 x+455 y &=994\end{aligned}\right.$$

Step-by-Step Solution

Answer

The lines intersect at approximately \((1.18, 0.56)\).

Step 1: Solve for y in the First Equation

The first equation is \(-435x + 912y = 0\). We need to solve for \(y\):Add \(435x\) to both sides: \[912y = 435x\].Now, divide every term by \(912\): \[y = \frac{435}{912}x\].Thus, the equation in terms of \(y\) is \(y = 0.477x\).

Step 2: Solve for y in the Second Equation

The second equation is \(132x + 455y = 994\). We need to solve for \(y\):Subtract \(132x\) from both sides:\[455y = 994 - 132x\].Now, divide every term by \(455\):\[y = \frac{994}{455} - \frac{132}{455}x\].Simplifying gives:\[y = 2.184 - 0.290x\].

Step 3: Graph the Equations

Plot the two equations using a graphing device:- The first line is \(y = 0.477x\).- The second line is \(y = 2.184 - 0.290x\).Ensure both lines are visible in the same viewing rectangle.

Step 4: Find the Intersection Point

Use the 'Intersect' feature, or zoom in and apply '[RACE]' on your graphing device to find where the two lines intersect. The intersection point represents the solution to the system of equations.

Step 5: Round the Intersection to Two Decimal Places

After identifying the intersection point, round the \(x\) and \(y\) values to two decimal places.

Key Concepts

Solving Linear Equations

When we talk about solving linear equations in a system, we are looking for a common solution to all given equations in the line. Linear equations typically look like this: \( ax + by = c \). Each letter represents a constant in your equation:

• \(a\) is the coefficient of \(x\)
• \(b\) is the coefficient of \(y\)
• \(c\) is a constant

To solve for a specific variable, you often rearrange the equation so that the variable is isolated. In our exercise, we rearranged the equations to solve for \(y\) in terms of \(x\):
- For the equation \(-435x + 912y = 0\), rearrange and simplify to get \(y = 0.477x\).
- For \(132x + 455y = 994\), rearrange and simplify to arrive at \(y = 2.184 - 0.290x\).
These simpler forms are what we use to plug into a graphing calculator easily.

Graphing Calculator

A graphing calculator is a handy tool that can display graphs of equations and find intersections. When using a graphing calculator:

• Enter each equation in its simplified form.
• Ensure both lines or equations appear clearly on the screen within the same viewing plot.
• You can use functionalities such as zoom or trace to pinpoint specific points on the graph.

In our exercise, once both equations \(y = 0.477x\) and \(y = 2.184 - 0.290x\) are graphed, you use the device to find the intersection where the two lines meet. This device significantly simplifies the process by giving a visual representation of your system of equations.

Intersection Point of Lines

The intersection point of lines is crucial because it represents the solution to a system of equations. On a graph, an intersection point is where two graphs meet. It gives us the value of \(x\) and \(y\) that satisfies both equations simultaneously.
To find this using a graphing calculator, employ the "Intersect" feature. This function identifies the exact coordinates where the lines cross.

• First, navigate through your graph using your calculator's navigation buttons.
• Once close to the intersection, apply the "Intersect" tool to find an approximate point of intersection.
• Adjust or zoom in further if needed for accuracy.

Finally, when you obtain the intersection's coordinates, round each component to two decimal places, ensuring your solutions are precise and easy to work with.