Identify the solution(s) of the system of equations, if any.
-3x -4y = 2
8y = -6x – 4
a) infinitely many solutions
b) (-16/9, 5/6)
c) (-2/3, 0)
d) no solution
8 y = -6 x - 4
8 y = -6 x - 4 the two equations are identical. Any of the infinite number of points on the line satisfies both.
a) infinitely many solutions
To solve this system of equations, we can use elimination or substitution. Let's use elimination:
1. Multiply both sides of the second equation by 2 to make the coefficient of x the same as in the first equation:
8y = -6x - 4 becomes 16y = -12x - 8.
2. Now we can multiply both sides of the first equation by -2:
-3x - 4y = 2 becomes 6x + 8y = -4.
3. Now we have two equations with the same coefficient for x:
6x + 8y = -4
16y = -12x - 8
4. Multiply the second equation by 3:
16y = -12x - 8 becomes 48y = -36x - 24.
5. Add the two equations together:
6x + 8y + 48y = -4 + (-36x - 24)
6x + 56y = -4 - 36x - 24
42x + 56y = -28
6. Divide both sides of the equation by 14 to simplify:
3x + 4y = -2
Now we have the following equation:
42x + 56y = -28
3x + 4y = -2
These two equations represent the same line. Therefore, they have infinitely many solutions.
So, the correct answer is a) infinitely many solutions.