Slope in Geometry Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy

Which of these lines is steeper:

y = 5x + 2

y = -7x - 3

? Which has a greater angle with the

x

-axis?

Solution

1
Step 1: Slopes: $m_1 = 5$ and $m_2 = -7$ . The absolute values are $|m_1| = 5$ and $|m_2| = 7$ .
2
Step 2: The steeper line has the larger $|m|$ , so $y = -7x - 3$ is steeper.
3
Step 3: Both $\theta_1 = \arctan(5) \approx 78.7°$ and $\theta_2 = 180° - \arctan(7) \approx 180° - 81.9° = 98.1°$ (obtuse, measured from positive $x$ -axis). By steepness of inclination, $|m_2| > |m_1|$ so $y = -7x-3$ makes a larger acute angle with the $x$ -axis.

Answer

y = -7x - 3

is steeper since

|-7| > |5|

Steepness is determined by the absolute value of the slope. A negative slope still produces a steep line if its magnitude is large. The angle a line makes with the