4 For each system, determine whether it has a unique solution (in this case, fin

Question

4 For each system, determine whether it has a unique solution (in this case, find the solution), infinitely many solutions, or no solutions 2 x +4y-24 4x+8y-48 Unique solution: x-0, y-o Unique solution: x-12, y-0 Unique solution: x-24, y-48 Infinitely many solutions No solutions 4x-5y 3x +2y0 Unique solution: x-0 y-o Infinitely many solutions Unique solution: x--2, y--4 No solutions Unique solution: x--9, y-5 6 x +2y20 18 x+6y59 Infinitely many solutions Unique solution: x-0, y 0 Unique solution: x--59, y--20 No solutions Unique solution: x--20, y-59 9 x-2y--64 4x +9y4 No solutions O Unique solution: x-0, y-o Unique solution: x--4,y-8 Infinitely many solutions Unique solution: x-8, y-4

Explanation / Answer

-2x + 4y = 24
-4x+8y = 48

If we see, these two equations are one and the same. Multiply the first equation by '2' and you get the second equation; Thus, these two equations effectively repreent the straight line -x + 2y = 12
Thus, there are infinitely many points on this straight line so there are infinitely many solutions

2) -4x - 5y=0
3x+2y=0;
3x=-2y so x= -2y/3
-4x - 5y=0
-4 (-2y/3) - 5y=0
8y/3-5y=0
8y/3 - 15y/3 =0
-7y/3=0 so y=0;
if y=0 then x= -2y/3= -2*0/3 = 0;
Thus, the unique solution is x=0, y=0

6x+2y = -20
18x + 6y = -59
The slope of these two lines is the same : -3 since -6/2 = -18/6 = -3;
But these two lines are different, so these are a pair of parallel lines and they will not intersect, hence no solution.

-9x-2y = -64 so 2y= 64-9x so y= (64-9x)/2
4x+9y = -4
4x + 9 (64-9x)/2 = -4
4x + 288 - 81x/2 = -4
-73x/2 = - 292
x= 292*2/73 = 8
y= 64-9x / 2 = 64-9*8 / 2 = (64-72)/2 = -8/2 = -4
Thus, a unique solution exists and that is x=-4, y=8;

Note: The key here is to identify that these equations represent a pair of straight lines. So find out if these are parallel lines, same lines or non-parallel but different lines that intersect at one point (point of solution)

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