The slope of a line is a measure of the amount the line rises or falls vertically for each unit of horizontal change. It's determined by the differences in the variables of two points on the line. Let's analyze each of these expressions to see which ones accurately represent the slope.
a. ∆y/∆x - This is the standard formula for slope, representing the change in y (the dependent variable) per unit change in x (the independent variable). This expression is accurately describing the slope.
b. run/rise - This expression is a variant of the slope formula, but it is technically incorrect. Slope is usually expressed as a ratio of the "rise" (the vertical change) to the "run" (the horizontal change), making the correct phrase "rise/run," not "run/rise."
c. x₂ - x₁/y₂ - y₁ - This formula is incorrect. The correct formula should be (y₂ - y₁) / (x₂ - x₁), where y₂ - y₁ represents the change in y (rise) and x₂ - x₁ represents the change in x (run).
d. change in the dependent variable relative to change in the independent variable - This statement is essentially describing the concept of slope, which is the rate of change of the dependent variable (usually y) with respect to the independent variable (usually x).
So, the correct expressions for slope are option a and option d.
Answer: d. change in the dependent variable relative to change in the independent variable