To determine the value of [tex]\( c \)[/tex] that makes the two equations [tex]\( 5x - 3y = 30 \)[/tex] and [tex]\( -10x + 6y = c \)[/tex] represent the same line, we need to investigate the forms of the equations.
First, observe the original equations:
1. [tex]\( 5x - 3y = 30 \)[/tex]
2. [tex]\( -10x + 6y = c \)[/tex]
To compare the two equations, let's try to manipulate the second equation so it resembles the first. Start by simplifying the second equation. Notice that each term in the second equation is a multiple of each term in the first equation:
[tex]\[ -10x + 6y = (-2) (5x - 3y) \][/tex]
Rewriting, we get:
[tex]\[ -10x + 6y = -2(5x - 3y) \][/tex]
Now distribute the [tex]\(-2\)[/tex]:
[tex]\[ -10x + 6y = -2 \cdot 5x + (-2) \cdot (-3y) \][/tex]
[tex]\[ -10x + 6y = -10x + 6y \][/tex]
This confirms the left side terms match up correctly. For the equations to be the same line, their right side constants must also match. Equate the right sides of the simplified version:
[tex]\[ -2 \times 30 = c \][/tex]
This results in:
[tex]\[ c = -60 \][/tex]
Therefore, the value of [tex]\( c \)[/tex] that makes the two given equations represent the same line is:
[tex]\[ \boxed{-60} \][/tex]
