Q22. - Algebra 2

Question

Compare the quantity in Column A and the quantity in Column B. Then determine whether:

A. the quantity in Column A is greater,

B. the quantity in Column B is greater,

C. the two quantities are equal, or 

D. the relationship cannot be determined from the information given.

 

Column A	Column B
The slope of line m	1

Step-by-Step Solution

Answer

The quantity in Column A is greater. Option A is correct.

Step-1 – Apply the concept of slope

The slope of the line passing through the points $\left(x_1,y_1\right)\text{ and }\left(x_2,y_2\right)$ is expressed below.

$m=\frac{y_2-y_1}{x_2-x_1}$

Step-2 –Substitute and solve

Consider the figure provided below.

Consider the ordered pairs $\left(1-k,0\right)\text{ and }\left(0,k\right)$.

Here $\left(x_1,y_1\right)\text{=}\left(1-k,0\right)\text{ and }\left(x_2,y_2\right)=\left(0,k\right)$.

Recall the formula of slope of the line and apply it.

$\begin{array}{c}m=\frac{y_2-y_1}{x_2-x_1}\\ =\frac{k-0}{0-\left(1-k\right)}\\ =\frac{k}{k-1}\end{array}$

Step-3 – Compare the columns A and B

The value in column B is 1. The value obtained in column A is $\frac{k}{k-1}$. Now, the value $\frac{k}{k-1}$ is always greater than 1 because numerator $k$ is greater than denominator $\left(k-1\right)$.

Hence, the quantity in Column A is greater.