Question

ms. blanco found that the slope of the line that passes through the points (-6, 1) and (2, 5) is 2. her work is below
$m = \frac{-6 - 2}{1 - 5} = \frac{-8}{-4} = 2$
select the correct response to represent ms. blanco’s work.
○ ms. blanco is correct
○ ms. blanco is incorrect, the slope should be -2. she forgot the negative sign.
○ ms. blanco is incorrect, the slope is $\frac{1}{2}$. she switched the x and y values in the formula.
○ ms. blanco is incorrect, the slope is $-\frac{3}{2}$. she did not follow the rule of subtraction.

Explanation:

Step1: Recall slope formula

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m=\frac{y_2 - y_1}{x_2 - x_1}\) (or \( \frac{y_1 - y_2}{x_1 - x_2}\), as long as consistency is maintained).

Step2: Apply formula to given points

Given points \((-6, 1)\) (so \( x_1=-6, y_1 = 1\)) and \((2, 5)\) (so \( x_2 = 2, y_2=5\)). Correct slope calculation: \( m=\frac{5 - 1}{2-(-6)}=\frac{4}{8}=\frac{1}{2}\)? Wait, no—wait, Ms. Blanco's work: she used \( m=\frac{-6 - 2}{1 - 5}\). Let's check her numerator and denominator. She took \( x_1 - x_2\) ( -6 - 2) for numerator and \( y_1 - y_2\) (1 - 5) for denominator. The slope formula is \( \frac{y_2 - y_1}{x_2 - x_1}\) or \( \frac{y_1 - y_2}{x_1 - x_2}\). So she switched x and y in the formula. Let's recalculate her work: \( \frac{-6 - 2}{1 - 5}=\frac{-8}{-4}=2\), but the correct formula should be \( \frac{y_2 - y_1}{x_2 - x_1}=\frac{5 - 1}{2 - (-6)}=\frac{4}{8}=\frac{1}{2}\)? Wait, no—wait, no, wait: if we use \( \frac{y_1 - y_2}{x_1 - x_2}\), that's \( \frac{1 - 5}{-6 - 2}=\frac{-4}{-8}=\frac{1}{2}\). But Ms. Blanco used \( \frac{x_1 - x_2}{y_1 - y_2}\), which is switching x and y. So her formula is wrong (switched x and y values in the formula). Let's check the options: the third option says "Ms. Blanco is incorrect, the slope is \( \frac{1}{2}\). She switched the x and y values in the formula." Let's verify: if we do \( \frac{y_2 - y_1}{x_2 - x_1}=\frac{5 - 1}{2 - (-6)}=\frac{4}{8}=\frac{1}{2}\). Or \( \frac{y_1 - y_2}{x_1 - x_2}=\frac{1 - 5}{-6 - 2}=\frac{-4}{-8}=\frac{1}{2}\). Ms. Blanco did \( \frac{x_1 - x_2}{y_1 - y_2}=\frac{-6 - 2}{1 - 5}=\frac{-8}{-4}=2\), which is wrong because she used x differences in numerator and y differences in denominator (switched x and y in the slope formula). So her mistake is switching x and y in the formula, and the correct slope is \( \frac{1}{2}\), which matches the third option.

Answer:

Ms. Blanco is incorrect, the slope is \(\frac{1}{2}\). She switched the x and y values in the formula. (The option corresponding to this statement, e.g., the third option in the list: "Ms. Blanco is incorrect, the slope is \(\boldsymbol{\frac{1}{2}}\). She switched the x and y values in the formula.")