To determine which equation has the steepest graph, we need to compare the slopes of the given linear equations. The slope of a line in the equation [tex]\( y = mx + b \)[/tex] is represented by the coefficient [tex]\( m \)[/tex]. Here are the given equations with their respective slopes:
A. [tex]\( y = 3x + 1 \)[/tex] (slope = 3)
B. [tex]\( y = 5x - 2 \)[/tex] (slope = 5)
C. [tex]\( y = -7x + 7 \)[/tex] (slope = -7)
D. [tex]\( y = \frac{1}{2}x + 3 \)[/tex] (slope = \frac{1}{2})
To find the steepest graph, we should consider the absolute values of the slopes because we are interested in the magnitude regardless of the direction (positive or negative).
Let's compare the absolute values of the slopes:
- The absolute value of the slope for equation A is [tex]\( |3| = 3 \)[/tex]
- The absolute value of the slope for equation B is [tex]\( |5| = 5 \)[/tex]
- The absolute value of the slope for equation C is [tex]\( |-7| = 7 \)[/tex]
- The absolute value of the slope for equation D is [tex]\( \left|\frac{1}{2}\right| = 0.5 \)[/tex]
Next, we compare these absolute values:
- 3
- 5
- 7
- 0.5
Among these values, 7 is the largest. Therefore, the equation with the steepest graph is the one with the slope that has the largest absolute value.
Thus, the equation with the steepest graph is:
C. [tex]\( y = -7x + 7 \)[/tex]
