Question

Use the following data to work Problems 15 to 17 . Draw a graph that shows the relationship between the two variables \(x\) and \(y\) $$\begin{array}{rrrrrrr} x & 0 & 1 & 2 & 3 & 4 & 5 \\ y & 25 & 24 & 22 & 18 & 12 & 0 \end{array}$$ Calculate the slope of the relationship between \(x\) and \(y\) when \(x\) equals 3

Short answer

The slope of the relationship between x and y when x equals 3 is -5.

Step-by-step solution

Understand the Data

The table provides pairs of values for the variables x and y. The objective is to plot these points on a graph and analyze the relationship between x and y.

Plot the Points on a Graph

On a Cartesian coordinate system, plot each pair \((x, y)\) from the table. The points to plot are: (0, 25), (1, 24), (2, 22), (3, 18), (4, 12), and (5, 0).

Draw the Line of Best Fit

Once the points are plotted, draw a line that best fits the points. This line will help in understanding the relationship between x and y.

Identify Points Around x = 3

To find the slope around x = 3, identify the points immediately surrounding it. Use the points (2, 22) and (4, 12) for calculation.

Use the Slope Formula

The slope (m) is calculated using the formula \[ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \] where (x1, y1) and (x2, y2) are the points (2, 22) and (4, 12) respectively.

Plug in the Values

Substitute the values into the formula: \[ m = \frac{{12 - 22}}{{4 - 2}} = \frac{{-10}}{{2}} = -5 \]

Interpret the Slope

The slope of -5 indicates that for every unit increase in x, y decreases by 5 units. This slope is relevant around the point where x = 3.

Key concepts

slope calculation

Understanding how to calculate the slope is key to analyzing linear relationships. The slope shows how one variable changes in relation to another. For the given data set, we use the points around the target x value (x=3), namely (2, 22) and (4, 12). You'll plug these into the slope formula \( m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \). Substituting the values, you get \( m = \frac{{12 - 22}}{{4 - 2}} = \frac{{-10}}{{2}} = -5 \). This negative slope tells us that as x increases, y decreases. Specifically, y drops by 5 for each unit increase in x. Knowing how to do this calculation helps in understanding how variables move together or apart.

plotting points

Plotting points helps visualize relationships in data. It starts by identifying pairs (x, y) from the table: (0, 25), (1, 24), (2, 22), (3, 18), (4, 12), and (5, 0). Place these on a Cartesian coordinate system: \[ \text{(x, y)} \] means \[ (0, 25) \] is at 0 on the x-axis and 25 on the y-axis. Similarly, plot the others: (1, 24), (2, 22), etc. Plotting makes it easy to see the trend or pattern in the relationship. Take your time with this step to ensure accuracy. Correct plotting sets the stage for drawing the line of best fit and further analysis.

line of best fit

A line of best fit best captures the trend of the plotted data points. This line may not pass through all points but should reflect the general pattern. To draw it, follow the data trend and balance the line among the points. Imagine it's balancing on the data points. It can help to use a ruler for straight lines. The line of best fit summarizes the relationship in a simple, visual way. When drawn, you can extend it to predict values outside your data range. It's a great way to estimate and make sense of the relationship.

analyzing data

Finally, analyzing the data helps in understanding the real-world relationship between variables. Here, the decrease in y values as x increases shows an inverse relationship. The negative slope highlights this. It means that as one variable goes up, the other goes down. Calculating the slope between selected points highlights this pattern clearly. Interpreting the slope helps make economic or other real-world predictions. For instance, in economics, this kind of relationship can apply to supply and demand, where one increases as the other decreases.