To determine which of the given linear equations has the steepest slope, we need to evaluate the magnitude (absolute value) of the slopes of each equation and compare them. Let's break it down step-by-step for each equation:
1. Equation A: [tex]\( y = \frac{2}{3}x + 4 \)[/tex]
- The slope is [tex]\( \frac{2}{3} \)[/tex].
- The absolute value of this slope is [tex]\( \left| \frac{2}{3} \right| = 0.6666666666666666 \)[/tex].
2. Equation B: [tex]\( y = -2x + 11 \)[/tex]
- The slope is [tex]\( -2 \)[/tex].
- The absolute value of this slope is [tex]\( \left| -2 \right| = 2 \)[/tex].
3. Equation C: [tex]\( y = \frac{1}{4}x + 7 \)[/tex]
- The slope is [tex]\( \frac{1}{4} \)[/tex].
- The absolute value of this slope is [tex]\( \left| \frac{1}{4} \right| = 0.25 \)[/tex].
4. Equation D: [tex]\( y = -7x + 2 \)[/tex]
- The slope is [tex]\( -7 \)[/tex].
- The absolute value of this slope is [tex]\( \left| -7 \right| = 7 \)[/tex].
Now, we compare the absolute values of the slopes:
- [tex]\( \left| \frac{2}{3} \right| = 0.6666666666666666 \)[/tex]
- [tex]\( \left| -2 \right| = 2 \)[/tex]
- [tex]\( \left| \frac{1}{4} \right| = 0.25 \)[/tex]
- [tex]\( \left| -7 \right| = 7 \)[/tex]
Among these, the largest absolute value is [tex]\( 7 \)[/tex]. Hence, the equation with the steepest slope is:
D. [tex]\( y = -7x + 2 \)[/tex]
