Answer :
To determine if the statement is true or false, let's consider the points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] on a line.
The slope [tex]\(m\)[/tex] of a line through two points is calculated using the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Now, let's examine the alternative way of labeling the points, which essentially reverses the order of the points:
[tex]\[ m = \frac{y_1 - y_2}{x_1 - x_2} \][/tex]
If we simplify this alternative slope formula:
[tex]\[ m = \frac{-(y_2 - y_1)}{-(x_2 - x_1)} \][/tex]
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
As you can see, reversing the order of the points essentially cancels out the negative signs both in the numerator and the denominator, leaving us with the original slope calculation:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Thus, regardless of the order in which we label the points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], the slope calculation ultimately yields the same result.
Therefore, the statement:
"It doesn't matter which of the two points on a line you choose to call [tex]\((x_1, y_1)\)[/tex] and which you choose to call [tex]\((x_2, y_2)\)[/tex] to calculate the slope of the line"
is:
A. True
The slope [tex]\(m\)[/tex] of a line through two points is calculated using the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Now, let's examine the alternative way of labeling the points, which essentially reverses the order of the points:
[tex]\[ m = \frac{y_1 - y_2}{x_1 - x_2} \][/tex]
If we simplify this alternative slope formula:
[tex]\[ m = \frac{-(y_2 - y_1)}{-(x_2 - x_1)} \][/tex]
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
As you can see, reversing the order of the points essentially cancels out the negative signs both in the numerator and the denominator, leaving us with the original slope calculation:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Thus, regardless of the order in which we label the points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], the slope calculation ultimately yields the same result.
Therefore, the statement:
"It doesn't matter which of the two points on a line you choose to call [tex]\((x_1, y_1)\)[/tex] and which you choose to call [tex]\((x_2, y_2)\)[/tex] to calculate the slope of the line"
is:
A. True
