Problem 52 - Algebra and Trigonometry

Question

\(49-52\) 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 [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 solution to the system, rounded to two decimal places, is approximately \((0.84, 0.40)\).

Step 1: Solve the First Equation for y

We begin by solving the first equation, \(-435x + 912y = 0\), for \(y\). Add \(435x\) to both sides to get: \[912y = 435x\]Next, divide both sides by 912 to isolate \(y\): \[y = \frac{435}{912}x\]Simplify \(\frac{435}{912}\) to get the slope-intercept form for the first equation.

Step 2: Solve the Second Equation for y

Proceed with the second equation, \(132x + 455y = 994\), and solve for \(y\). Subtract \(132x\) from both sides to yield:\[455y = -132x + 994\]Then, divide all terms by 455:\[y = -\frac{132}{455}x + \frac{994}{455}\]This form is ready for graphing.

Step 3: Graph the Lines

Using a graphing device, input both equations. The first equation is \(y = \frac{435}{912}x\) and the second equation is \(y = -\frac{132}{455}x + \frac{994}{455}\). Graph them in the same coordinate plane to visually identify their intersection.

Step 4: Zoom In on the Intersection

Use the zoom functionality of the graphing tool to closely examine where the two lines cross. The accuracy needed requires zooming in such that the intersection is clear and precise.

Step 5: Use TRACE or Intersect Feature

Utilize the TRACE feature to move along the graph and identify approximate coordinates of the intersection. Alternatively, use the Intersect feature on many graphing calculators to compute the exact intersection point.

Step 6: Round the Intersection Point

Once the intersection point \((x, y)\) is identified, round both the x-coordinate and y-coordinate to two decimal places. This yields the solution to the system.

Key Concepts

Graphing Linear Equations

Graphing linear equations is a fundamental skill in algebra that allows us to visually explore the relationship between variables. A linear equation, typically expressed in the form of \( y = mx + b \), consists of a straight line when graphed on a coordinate plane. The "\(m\)" denotes the slope of the line, which represents how steep the line is, while "\(b\)" indicates the y-intercept, or where the line crosses the y-axis.

• First, solve the equation for \(y\) in terms of \(x\).
• Identify the slope and y-intercept to plot the line.
• Use a graphing device or graphing calculator to insert these equations.
• Observe the plotted line on the coordinate plane.

This process helps us see where the line will intersect other lines, which is crucial when solving systems of equations by graphing.

Intersection of Lines

The intersection of lines refers to the point where two lines on a graph cross each other. This point represents the solution to a system of linear equations, meaning it satisfies both equations at once.

Finding where lines intersect involves graphing each equation in the system and identifying the exact spot they meet.

• If the lines intersect at a single point, this means the system has a unique solution.
• If the lines are parallel and never intersect, the system has no solution.
• If the lines lie on top of each other, meaning they coincide, there are infinitely many solutions.

Often, a graphing calculator's "Intersect" feature can be used to precisely locate the intersection, providing accurate coordinates of this vital point. This approach offers a clear and visual method to find solutions, especially helpful in complex systems.

Solving for y in Terms of x

Solving for \(y\) in terms of \(x\) is a critical step in preparing linear equations for graphing, especially when using graphing calculators. It involves manipulating the equation to isolate \(y\) on one side, resulting in a slope-intercept form such as \( y = mx + b \).

To solve for \(y\):

• Rearrange the equation, moving all terms except \(y\) to the opposite side.
• Perform arithmetic operations to leave \(y\) by itself.
• Simplify any fractions if possible to make graphing easier.

For example, in the equation \(-435x + 912y = 0\), rearrange and solve, leading to \(y = \frac{435}{912}x\).

This method ensures that the line is correctly graphed, with \(m\) as the slope, detailing how the line rises or falls, and \(b\) showing where it crosses the y-axis. This format is crucial for visual comparison of different lines.