Problem 112 - College Algebra

Question

For the following exercises, use the functions \(f(x)=-0.1 x+200\) and \(g(x)=20 x+0.1\). Find the point of intersection of the lines \(f\) and \(g\).

Step-by-Step Solution

Answer

The lines intersect at approximately (9.95, 199.005).

Step 1: Set the Functions Equal

To find the point of intersection between two lines, set the functions equal to each other: \[-0.1x + 200 = 20x + 0.1\].

Step 2: Solve for x

Rearrange the equation to solve for \(x\). This involves moving all terms involving \(x\) to one side of the equation and constant terms to the other side:\[-0.1x - 20x = 0.1 - 200\]Combine like terms:\[-20.1x = -199.9\]Now, divide by \(-20.1\) to solve for \(x\):\[x = \frac{-199.9}{-20.1}\]Calculate the value of \(x\):\[x \approx 9.95\].

Step 3: Solve for y

Use either \(f(x)\) or \(g(x)\) to find the corresponding \(y\) value. We will use \(f(x)\):\[f(9.95) = -0.1(9.95) + 200\]Calculate:\[f(9.95) \approx 199.005\]Thus, \(y \approx 199.005\).

Step 4: State the Point of Intersection

Write down the coordinates of the point of intersection:The point of intersection is approximately \((9.95, 199.005)\).

Key Concepts

Linear Functions

Linear functions are mathematical expressions that graph as straight lines in the coordinate plane. They are called linear because of their form, which is a straight line when plotted. A linear function can generally be expressed as \( y = mx + b \), where:

• \( m \) represents the slope of the line, describing how steep the line is.
• \( b \) is the y-intercept, the point at which the line crosses the y-axis.

Given two linear functions in the problem, \( f(x)=-0.1x+200 \) and \( g(x)=20x+0.1 \), you can see:

• The function \( f(x) \) has a slope of -0.1 and a y-intercept of 200.
• The function \( g(x) \) has a slope of 20 and a y-intercept of 0.1.

In the context of linear functions, understanding the slope is crucial. It tells us the direction and rate of change of the line. The y-intercept tells us where the line starts on the y-axis. Knowing these two functions intersecting is a fascinating feature of linear equations. When graphed, their intersection point is where they "meet" on the graph.

Intersection Points

The intersection point between two lines is a particular point where both lines cross each other on the coordinate plane. It signifies a solution where both equations hold true. To find this point, you must find the values of \( x \) and \( y \) that satisfy both equations simultaneously.
One way to find intersection points is by setting the equations of the two lines equal, which is exactly what we did in the problem: \[-0.1x + 200 = 20x + 0.1\]
This step essentially tells us to find common ground — the exact point where both lines share the same coordinates. Solving for \( x \) determines the specific value at the intersection horizontally. Once \( x \) is found, substituting it back into either original equation will then give the \( y \) coordinate, completing the intersection point. This method showcases an elegant interplay between algebra and geometry, emphasizing the utility and power of algebraic manipulation in understanding spatial relationships.

Solving Equations

Solving equations is the process of finding the unknown variable that makes the equation true. For finding the intersection of lines, this involves manipulating the given equations to isolate the variable.

• First, we align like terms on either side of the equation.
• Next, combine like terms to simplify.
• Then, solve for the unknown variable through division or other arithmetic operations.

In our exercise, once the equations were set equal: \[-0.1x - 20x = 0.1 - 200\]
We combined terms to isolate \( x \): \[-20.1x = -199.9\]
Dividing both sides by \(-20.1\) gave us \( x \approx 9.95 \).
Solving for \( x \) using these steps is crucial in algebra as it applies not only to finding intersection points but also to many other types of problems. Understanding this process empowers students to handle various equation-solving scenarios, providing a solid foundation for more advanced mathematical concepts.