Let’s solve the problem step by step to determine which two equations are equivalent.

Given equations:

1. $y - 8 = \frac{2}{5}(x - 4)$
2. $2x - 3y - 32 = 0$
3. $y = \frac{2}{3}x + \frac{16}{3}$

Step 1: Rewrite equation 1 in slope-intercept form ($y = mx + b$):

The equation is: $y - 8 = \frac{2}{5}(x - 4)$

Distribute $\frac{2}{5}$: $y - 8 = \frac{2}{5}x - \frac{8}{5}$

Add 8 to both sides: $y = \frac{2}{5}x - \frac{8}{5} + 8$

Convert 8 to a fraction with denominator 5: $y = \frac{2}{5}x - \frac{8}{5} + \frac{40}{5}$

Simplify: $y = \frac{2}{5}x + \frac{32}{5}$

Step 2: Rewrite equation 2 in slope-intercept form:

The equation is: $2x - 3y - 32 = 0$

Rearrange for $y$: $-3y = -2x + 32$

Divide by $-3$: $y = \frac{2}{3}x - \frac{32}{3}$

Step 3: Compare equation 3 ($y = \frac{2}{3}x + \frac{16}{3}$):

We already have the slope-intercept forms:

• From equation 1: $y = \frac{2}{5}x + \frac{32}{5}$
• From equation 2: $y = \frac{2}{3}x - \frac{32}{3}$
• Equation 3 is $y = \frac{2}{3}x + \frac{16}{3}$.

The slopes ($m$) and y-intercepts ($b$) are not the same across all three forms, so none of these equations are identical.

Conclusion:

The equivalent equations are equation 2 and equation 3, as they both have the same slope ($\frac{2}{3}$) but different y-intercepts. Would you like a visual breakdown of these steps?