Find a linear model for the above data using the points (0,344.3) and (22,380.9).
The correct answer and explanation is:
To find a linear model based on the two points:
(0, 344.3) and (22, 380.9),
we will follow the standard form of a linear equation: y=mx+by = mx + b
Where:
- mm is the slope,
- bb is the y-intercept.
Step 1: Find the slope (m)
The slope formula between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is: m=y2−y1x2−x1m = \frac{y_2 – y_1}{x_2 – x_1}
Substitute the values: m=380.9−344.322−0=36.622≈1.6636m = \frac{380.9 – 344.3}{22 – 0} = \frac{36.6}{22} \approx 1.6636
Step 2: Find the y-intercept (b)
We use one point (e.g., (0, 344.3)) and the slope to solve for bb: y=mx+b⇒344.3=(1.6636)(0)+b⇒b=344.3y = mx + b \Rightarrow 344.3 = (1.6636)(0) + b \Rightarrow b = 344.3
Final Linear Model
y=1.6636x+344.3y = 1.6636x + 344.3
Explanation (300 words)
A linear model helps describe a straight-line relationship between two variables. In this case, the data is represented by two specific points: (0, 344.3) and (22, 380.9). These points can be interpreted as coordinates on a graph where the x-values represent an independent variable (like time, years, or quantity), and the y-values represent a dependent variable (like cost, temperature, or population).
To build a linear model, we first calculated the slope (rate of change) using the difference in y-values over the difference in x-values. This gives us how much the dependent variable increases for every one-unit increase in the independent variable. Here, the slope m=1.6636m = 1.6636 tells us that for each increase of 1 in x, y increases by about 1.66.
Next, we found the y-intercept (b), which is the value of y when x = 0. From the point (0, 344.3), it is clear that the y-intercept is 344.3. This serves as the starting value of the model when the independent variable is zero.
Therefore, the linear equation y=1.6636x+344.3y = 1.6636x + 344.3 allows us to predict values of y for any x. This type of model is commonly used in forecasting, business, science, and economics because it simplifies complex relationships into manageable, predictable forms.