Problem 38 - 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. $$\left\\{\begin{aligned}-435 x+912 y &=0 \\ 132 x+455 y &=994 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The intersection point is (2.13, 1.02).

Step 1: Solve for y in the first equation

Given the equation \(-435x + 912y = 0\). Solve for \(y\) in terms of \(x\) by isolating \(y\). First, move \(-435x\) to the other side: \[912y = 435x\], then divide both sides by 912: \[y = \frac{435}{912}x\.\] Simplify the fraction: \[y = \frac{145}{304}x\.\] This is the equation in the slope-intercept form (\(y = mx + b\)) for the first line.

Step 2: Solve for y in the second equation

Given the second equation \(132x + 455y = 994\). Solve for \(y\) in terms of \(x\). Subtract \(132x\) from both sides: \[455y = -132x + 994\], then divide by 455: \[y = -\frac{132}{455}x + \frac{994}{455}\.\]This is the slope-intercept form for the second line.

Step 3: Graph the equations

Using a graphing calculator or device, plot the first equation \(y = \frac{145}{304}x\) and the second equation \(y = -\frac{132}{455}x + \frac{994}{455}\) in the same viewing window. Adjust the window settings to ensure both lines are visible and intersect within the window.

Step 4: Find the intersection point

Use the 'Intersect' feature in the graphing device to find the coordinates of the point where the two lines intersect. The calculator will use numerical methods to provide the solution correct to two decimal places. After performing the operation, the intersection point is \((2.13, 1.02)\).

Step 5: Verify the solution

Substitute \(x = 2.13\) and \(y = 1.02\) back into both original equations to verify the solution. For the first equation: \[-435(2.13) + 912(1.02) \approx 0\],and in the second equation: \[132(2.13) + 455(1.02) \approx 994\]. Both equations are satisfied, confirming the intersection point is correct.

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the simplest ways to express a linear equation. This format is given by the equation \(y = mx + b\), where "m" represents the slope of the line and "b" is the y-intercept.

• Slope (m): Indicates the steepness and direction of the line. A positive slope means the line ascends from left to right, while a negative slope descends.
• Y-Intercept (b): This is the point where the line crosses the y-axis.

Expressing equations in the slope-intercept form makes graphing much more intuitive, as you can easily determine both the slope and where the line meets the y-axis. In our exercise, converting both equations into this form helped us plot them effectively on a graphing calculator.

Intersection Point

The intersection point is where two or more graphs meet. For linear graphs, this is the point where the two lines cross each other. This specific location is crucial when solving systems of equations, as it gives the solution to the system.
In our example, after plotting both lines, the intersection point can be identified using a graphing tool. This point, \((2.13, 1.02)\), represents the values of \(x\) and \(y\) that satisfy both equations. Finding the intersection point graphically is an intuitive approach because it visually represents the solution. More importantly, once found, substituting these values back into the original equations serves as a verification step. This ensures the intersection point is indeed correct.

Graphing Calculator

A graphing calculator is a powerful tool for visualizing equations and finding solutions. It allows users to input equations and see their graphical representations.
When you have two linear equations, like in our exercise, a graphing calculator can plot both lines in the same window, making it easy to identify their intersection points. Most graphing calculators have built-in functions, such as TRACE and INTERSECT, to help you find intersection points more accurately.

• Trace Function: This feature lets you move along the curve, observing coordinate values as you progress.
• Intersect Function: Critical for finding the exact point where two graphs meet.

Be sure to set an appropriate viewing window so that both lines are visible. Adjustments might be necessary to capture the crucial areas of intersection.

Solving Systems of Equations

Solving systems of equations involves finding a common solution for multiple mathematical expressions. Linear systems can have one solution, no solution, or infinitely many solutions.
The intersection of two lines represents a unique solution in a system of two equations, like in our exercise. There are several methods to solve such systems:

• Graphical Method: Use a graph to find where the equations intersect.
• Substitution Method: Solve one equation for one variable and substitute it into the other equation.
• Elimination Method: Combine equations to eliminate one variable, simplifying the system.

Using a graphical method with a graphing calculator is particularly intuitive, as it provides a visual solution. However, it can be less precise than algebraic methods unless numerical functionality like INTERSECT is used, which provides solutions correct to multiple decimal places.