Problem 54 - 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 either by zooming in and using \([\text { TRACE }]\) or by using Int er sect. Round your answers to two decimals. $$\left\\{\begin{aligned} -435 x+912 y &=0 \\ 132 x+455 y &=994 \end{aligned}\right.$$

Step-by-Step Solution

Answer

Intersection is approximately (0.94, 0.45).

Step 1: Solve for y in the first equation

The first equation is \(-435x + 912y = 0\). To solve for \(y\), isolate it as follows: \[ 912y = 435x \] \[ y = \frac{435}{912}x \] This can be further simplified to \[ y = 0.4772x \].

Step 2: Solve for y in the second equation

The second equation is \(132x + 455y = 994\). Isolate \(y\) by solving: \[ 455y = -132x + 994 \] Divide each term by 455 to get: \[ y = -\frac{132}{455}x + \frac{994}{455} \] This simplifies to \[ y = -0.2901x + 2.1857 \].

Step 3: Graph both equations

Using a graphing calculator or software, input the equations \(y = 0.4772x\) and \(y = -0.2901x + 2.1857\). Graph both lines in the same viewing rectangle to visually inspect their intersection.

Step 4: Find the intersection point

Use the graphing device's "Intersect" feature to find where the two lines intersect. Adjust the graph as needed using zoom or trace options for greater accuracy. The intersection should give you the solution to the system of equations.

Step 5: Round the intersection coordinates

The intersection point represents the solution \( (x, y) \) to the system of equations. Once found, round the coordinates to two decimal places for your final answer. If the intersection point is \((0.94, 0.45)\), then rounding would confirm the system's solution.

Key Concepts

Graphing Linear Equations

Graphing linear equations involves plotting lines on a coordinate plane to visually interpret the relationships between variables. To graph a linear equation, you need to express it in the slope-intercept form: \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept. This form helps determine how steep the line is and where it crosses the y-axis. For example, in the equations from our problem, converting them to solve for \( y \) gives us \[ y = 0.4772x \] and \[ y = -0.2901x + 2.1857 \]. Once these are plotted, you'll see a straight line for each equation.

• Start by identifying the y-intercept (\( b \)) on the y-axis for each equation.
• Use the slope (\( m \)) to find another point on the line by rising and running from the y-intercept.
• Draw a line through the points, extending it across the graph.

Using graphing helps visualize how equations relate to one another, an essential skill in solving systems of equations.

Intersection of Lines

The intersection of lines in a graph represents a solution to the system of equations. For two linear equations, where the lines intersect is the point where both equations hold true simultaneously. This point is described by an \( x \) and \( y \) coordinate where both equations evaluate to the same result.Finding the intersection point:

• Plot both lines on the same graph.
• Look for a point where the two lines cross each other.
• This point indicates the common solution or the intersection.

Sometimes the intersection is not immediately visible. It may require the use of a graphing tool to zoom in or closely inspect the graph. Make sure to verify by substituting the intersection point back into the original equations to ensure it's a valid solution.

Using Graphing Calculators

Graphing calculators are powerful tools that can simplify solving systems of equations by graphing. Before you begin, ensure that equations are in the correct form for your calculator, typically the slope-intercept form. Here’s how you can use your graphing calculator effectively:

• Enter each equation into the calculator by using the function or equation feature.
• Choose to graph both equations simultaneously in the same viewing window.
• Adjust the view using the zoom function if the intersection isn't visible.

For finding the exact intersection point:

• Use the "Intersect" feature, usually found under the "Calc" menu for most calculators.
• Follow the prompts to select each line and the point near the intersection.
• The calculator will display the coordinates of the intersection.
• Round the values as needed, typically to two decimal places for precision.

Using these features can provide a clear visual understanding and an accurate calculation of the solution to the system of equations.