To determine which equations also represent the given line [tex]\( y = -\frac{5}{3}(x-2) \)[/tex], let's analyze and compare each option.
Step-by-Step Solution:
1. Given Line Equation:
[tex]\[ y = -\frac{5}{3}(x - 2) \][/tex]
Let's expand this equation to see its standard form:
[tex]\[ y = -\frac{5}{3} x + \frac{10}{3} \][/tex]
2. Option 1: [tex]\( y = -\frac{5}{3}x - 2 \)[/tex]
Compare this with the given line's standard form [tex]\( y = -\frac{5}{3} x + \frac{10}{3} \)[/tex]:
[tex]\[ y \quad = \quad -\frac{5}{3}x \quad - 2 \][/tex]
This equation does not match [tex]\( y = -\frac{5}{3}x + \frac{10}{3} \)[/tex].
3. Option 2: [tex]\( y = -\frac{5}{3}x + \frac{10}{3} \)[/tex]
Compare this with the given line's standard form:
[tex]\[ y \quad = \quad -\frac{5}{3}x \quad + \frac{10}{3} \][/tex]
This equation matches [tex]\( y = -\frac{5}{3}x + \frac{10}{3} \)[/tex].
4. Option 3: [tex]\( 3y = -5x + 10 \)[/tex]
Rearrange to standard form [tex]\( y = mx + b \)[/tex]:
[tex]\[ 3y = -5x + 10 \][/tex]
Divide both sides by 3:
[tex]\[ y = -\frac{5}{3}x + \frac{10}{3} \][/tex]
This equation matches [tex]\( y = -\frac{5}{3}x + \frac{10}{3} \)[/tex].
5. Option 4: [tex]\( 3x + 15y = 30 \)[/tex]
Rearrange to standard form [tex]\( y = mx + b \)[/tex]:
[tex]\[ 15y = -3x + 30 \][/tex]
Divide both sides by 15:
[tex]\[ y = -\frac{1}{5}x + 2 \][/tex]
This equation does not match [tex]\( y = -\frac{5}{3}x + \frac{10}{3} \)[/tex].
6. Option 5: [tex]\( 5x + 3y = 10 \)[/tex]
Rearrange to standard form [tex]\( y = mx + b \)[/tex]:
[tex]\[ 3y = -5x + 10 \][/tex]
Divide both sides by 3:
[tex]\[ y = -\frac{5}{3}x + \frac{10}{3} \][/tex]
This equation matches [tex]\( y = -\frac{5}{3}x + \frac{10}{3} \)[/tex].
Hence, the equations that also represent the given line [tex]\( y = -\frac{5}{3} (x - 2) \)[/tex] are:
[tex]\[ \boxed{2, 3, 5} \][/tex]
