To determine the meaning of the [tex]$x$[/tex]-value on the line when [tex]$y=12$[/tex] in the context of Justin's research, let's go through the process step-by-step:
1. Understanding the Data Relationship:
- Justin is interested in understanding the relationship between the length (in inches) and the weight (assumed to be in a certain unit like pounds or ounces) of a species of catfish.
- To do this, he measures the lengths and weights of a number of catfish specimens and plots these measurements on a graph.
2. Plotting and the Line of Best Fit:
- Upon plotting the lengths (independent variable, [tex]$x$[/tex]) on the x-axis and the weights (dependent variable, [tex]$y$[/tex]) on the y-axis, Justin draws a line of best fit.
- The line of best fit is a straight line that best represents the data points on the plot, minimizing the distance between the actual data points and the line itself.
3. Forming the Equation:
- Let’s denote the equation of this line of best fit as [tex]\( y = a \cdot x + b \)[/tex], where:
- [tex]\( y \)[/tex] represents the weight of the catfish.
- [tex]\( x \)[/tex] represents the length of the catfish.
- [tex]\( a \)[/tex] is the slope of the line.
- [tex]\( b \)[/tex] is the y-intercept of the line.
4. Finding the Meaning of the x-value When y=12:
- We are asked to find the meaning of the [tex]$x$[/tex]-value when [tex]$y = 12$[/tex].
- By substituting [tex]\( y = 12 \)[/tex] into the linear equation [tex]\( y = a \cdot x + b \)[/tex], we get:
[tex]\[
12 = a \cdot x + b
\][/tex]
- Solving for [tex]\( x \)[/tex] would give us:
[tex]\[
x = \frac{12 - b}{a}
\][/tex]
5. Interpreting the Result:
- The [tex]$x$[/tex]-value obtained here represents a specific length of the catfish.
- Therefore, the meaning of the [tex]$x$[/tex]-value on the line when [tex]$y = 12$[/tex] is the length in inches of the catfish at which its weight is 12 units (which could be pounds, ounces, etc., based on the data Justin collected).
In summary, the [tex]$x$[/tex]-value on the line when [tex]$y=12$[/tex] represents the length in inches of the catfish at which its weight is 12 units. This interpretation helps us understand the particular length of a catfish that corresponds to a weight of 12 units according to the relationship established by the line of best fit from Justin's data.
