3. Which two quadrants contain all of the solutions to the following system?
y4x-2
y-3x+5
A. I and II
B. II and III
C. III and IV
D. I and IV
To determine which quadrants contain all of the solutions to the system, we need to find the intersecting point of the two lines.
To find the intersecting point, we can set the equations equal to each other:
4x - 2 = -3x + 5
Adding 3x and subtracting 2 from both sides:
7x = 7
Dividing both sides by 7:
x = 1
Substituting this value of x back into one of the original equations, we can find the value of y:
y = 4(1) - 2
y = 4 - 2
y = 2
So the intersecting point of the two lines is (1, 2).
Now we can determine which quadrants contain this point.
In Quadrant I, both x and y are positive. Since our point has positive values for both x and y (x = 1, y = 2), it lies in Quadrant I.
In Quadrant II, x is negative and y is positive. Since our point has a positive value for y (y = 2), but a positive value for x (x = 1), it does not lie in Quadrant II.
In Quadrant III, both x and y are negative. Since our point has positive values for both x and y (x = 1, y = 2), it does not lie in Quadrant III.
In Quadrant IV, x is positive and y is negative. Since our point has a positive value for x (x = 1), but a positive value for y (y = 2), it does not lie in Quadrant IV.
Therefore, the two quadrants that contain all of the solutions to the system are Quadrants I and II.
The correct answer is A. I and II.