To determine which of the given linear equations has the steepest slope, we need to look at the coefficients of [tex]\( x \)[/tex] in each equation. The coefficient of [tex]\( x \)[/tex] in a linear equation of the form [tex]\( y = mx + b \)[/tex] represents the slope.
Let's identify the slopes for each equation:
1. For equation A: [tex]\( y = \frac{2}{3}x + 10 \)[/tex]
- The slope is [tex]\( \frac{2}{3} \)[/tex].
2. For equation B: [tex]\( y = -4x + 6 \)[/tex]
- The slope is [tex]\( -4 \)[/tex].
3. For equation C: [tex]\( y = \frac{1}{7}x - 5 \)[/tex]
- The slope is [tex]\( \frac{1}{7} \)[/tex].
4. For equation D: [tex]\( y = 8x + 1 \)[/tex]
- The slope is [tex]\( 8 \)[/tex].
To determine which slope is the steepest, we compare the absolute values of these slopes, because the steepest slope has the greatest absolute value.
- [tex]\( |\frac{2}{3}| = 0.67 \approx 0.67 \)[/tex]
- [tex]\( |-4| = 4 \)[/tex]
- [tex]\( |\frac{1}{7}| = 0.143 \approx 0.14 \)[/tex]
- [tex]\( |8| = 8 \)[/tex]
Among these values, the greatest absolute value is [tex]\( 8 \)[/tex].
Therefore, the equation with the steepest slope is:
[tex]\[ y = 8x + 1 \][/tex]
Thus, the answer is D.
