Problem 88 - Precalculus

Question

Apply a graphing utility to graph the two equations \(y=14.76 x+19.43\) and \(y=2.76 x+5.22 .\) Approximate the solution to this system of linear equations.

Step-by-Step Solution

Answer

The solution to the system is the intersection point, approximately \((x, y) \approx (-1.13, 2.08)\).

Step 1: Introduce the Graphing Task

We need to use a graphing utility to visualize the straight lines represented by the equations \(y = 14.76x + 19.43\) and \(y = 2.76x + 5.22\). Our goal is to find the point where these two lines intersect, which will give us the solution to the system of equations.

Step 2: Graph the First Equation

Start by graphing the first equation \(y = 14.76x + 19.43\). This line has a slope of 14.76, meaning it rises steeply. The y-intercept is 19.43, indicating where the line crosses the y-axis.

Step 3: Graph the Second Equation

Next, plot the second equation \(y = 2.76x + 5.22\). This line has a milder slope of 2.76 and crosses the y-axis at 5.22.

Step 4: Find the Intersection Point

Observe where the two lines intersect on the graph. The intersection point represents the solution to the system of equations; it is the values of \(x\) and \(y\) that satisfy both equations simultaneously.

Step 5: Approximate the Intersection

Upon graphing the equations, utilize the graphing utility's feature to pinpoint the intersection. The graphing calculator might show this intersection as approximately \((x, y) \approx (x_0, y_0)\). Record these values as the solution.

Key Concepts

Intersection Point

In a system of linear equations, the intersection point is a vital concept. It is the point where two lines on a graph meet. By finding this point, you identify the values for both variables that satisfy each equation simultaneously. When considering the system of equations given—

• \(y = 14.76x + 19.43\)
• \(y = 2.76x + 5.22\)

The lines represented by these equations intersect at a unique point, indicating that both equations are true at those particular x and y values. Visualizing this graphically can give you a precise picture of where this intersection occurs. Plotting both lines on a graph reveals the intersection, allowing you to estimate the most accurate x and y solutions.

Slope

The slope of a line is an essential concept for understanding how steep the line is. It is the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. When you look at the equations:

• \(y = 14.76x + 19.43\)
• \(y = 2.76x + 5.22\)

These slopes are 14.76 and 2.76, respectively. A larger slope value like 14.76 means the line rises steeply. Conversely, a smaller slope of 2.76 indicates a gentler ascent. Comparing these slopes helps understand their intersection since the steepness affects where and how they meet. A greater slope will rapidly increase or decrease, while a smaller one progresses gradually. This difference plays a crucial role in determining the intersection point.

y-intercept

The y-intercept provides crucial information about where a line crosses the y-axis. It indicates the value of y when x is zero. In our equations, the y-intercepts are 19.43 and 5.22. Here's what that means:

• The first line crosses the y-axis at 19.43.
• The second line crosses the y-axis at 5.22.

This information allows us to quickly plot each line's starting point on a graph. When combined with the slope, you get a full picture of the line's position and direction on the graph. The y-intercept, along with the slope, is used to draw the line accurately and helps you understand the overall graph layout before finding the intersection.

Graphing Utility

A graphing utility is a tool that simplifies the process of plotting equations by visually representing them on a coordinate plane. These can be calculators, software, or apps that allow you to enter equations and see their graphs. In this exercise, a graphing utility is used to plot the lines of both equations:

• \(y = 14.76x + 19.43\)
• \(y = 2.76x + 5.22\)

The utility accurately maps these lines using the slope and y-intercept information. By doing so, it graphically demonstrates where these lines meet, hence finding the intersection point. Additionally, many graphing utilities have features that calculate the intersection precisely, offering a straightforward way to determine where the lines cross, enhancing understanding and providing exact solution values.