Answer :
To determine which expressions for slope are incorrect, let's analyze each option one by one:
- Option A: Change in the dependent variable relative to change in the independent variable
This expression correctly describes the slope. The slope is defined as the rate of change of the dependent variable (usually [tex]\( y \)[/tex]) relative to the change in the independent variable (usually [tex]\( x \)[/tex]). Hence, this is a correct expression for slope.
- Option B: Rise
The term "rise" alone is not a complete representation of the slope. Slope is defined as "rise over run" or the vertical change divided by the horizontal change. Therefore, using just "rise" is an incorrect expression for slope.
- Option C: [tex]\(\frac{\Delta y}{\Delta x}\)[/tex]
This is the standard mathematical formula for calculating the slope, where [tex]\(\Delta y\)[/tex] represents the change in the [tex]\( y \)[/tex] values and [tex]\(\Delta x\)[/tex] represents the change in the [tex]\( x \)[/tex] values. This is a correct expression for slope.
- Option D: [tex]\(\frac{x_2 - x_1}{y_2 - y_1}\)[/tex]
This expression is incorrect because the proper way to express slope is [tex]\(\frac{y_2 - y_1}{x_2 - x_1}\)[/tex]. The numerator and the denominator are swapped here, so this does not correctly describe the slope.
Based on the analysis:
- Option A: Correct (not an incorrect expression for slope)
- Option B: Incorrect (incomplete representation of slope)
- Option C: Correct (not an incorrect expression for slope)
- Option D: Incorrect (wrong mathematical expression for slope)
Thus, the incorrect expressions for slope are options B and D.
- Option A: Change in the dependent variable relative to change in the independent variable
This expression correctly describes the slope. The slope is defined as the rate of change of the dependent variable (usually [tex]\( y \)[/tex]) relative to the change in the independent variable (usually [tex]\( x \)[/tex]). Hence, this is a correct expression for slope.
- Option B: Rise
The term "rise" alone is not a complete representation of the slope. Slope is defined as "rise over run" or the vertical change divided by the horizontal change. Therefore, using just "rise" is an incorrect expression for slope.
- Option C: [tex]\(\frac{\Delta y}{\Delta x}\)[/tex]
This is the standard mathematical formula for calculating the slope, where [tex]\(\Delta y\)[/tex] represents the change in the [tex]\( y \)[/tex] values and [tex]\(\Delta x\)[/tex] represents the change in the [tex]\( x \)[/tex] values. This is a correct expression for slope.
- Option D: [tex]\(\frac{x_2 - x_1}{y_2 - y_1}\)[/tex]
This expression is incorrect because the proper way to express slope is [tex]\(\frac{y_2 - y_1}{x_2 - x_1}\)[/tex]. The numerator and the denominator are swapped here, so this does not correctly describe the slope.
Based on the analysis:
- Option A: Correct (not an incorrect expression for slope)
- Option B: Incorrect (incomplete representation of slope)
- Option C: Correct (not an incorrect expression for slope)
- Option D: Incorrect (wrong mathematical expression for slope)
Thus, the incorrect expressions for slope are options B and D.
