Solve the following system of equations: 3x – 2y = 6 6x – 4y = 12 A. (0,0) B. (6,12) C. Infinitely many solutions D. No solutions
The Correct Answer and Explanation is:
To solve the system of equations:
- 3x−2y=63x – 2y = 63x−2y=6
- 6x−4y=126x – 4y = 126x−4y=12
Begin by examining the relationship between the two equations. Multiply the first equation by 2: 2(3x−2y)=2(6)2(3x – 2y) = 2(6)2(3x−2y)=2(6) 6x−4y=126x – 4y = 126x−4y=12
This result is identical to the second equation: 6x−4y=126x – 4y = 126x−4y=12
Since both equations represent the same line, every solution of one equation is also a solution of the other. This means the system does not define two distinct lines intersecting at a single point or being parallel with no intersection. Instead, it represents the same line repeated twice.
Therefore, the system has infinitely many solutions. Any point (x,y)(x, y)(x,y) that satisfies the first equation also satisfies the second, and vice versa. In graphical terms, this means the two lines lie directly on top of one another, covering the same set of points on the coordinate plane.
To further verify this, consider expressing the solution in parametric form. Solve the first equation for yyy: 3x−2y=6−2y=−3x+6y=3x−623x – 2y = 6 \\ -2y = -3x + 6 \\ y = \frac{3x – 6}{2}3x−2y=6−2y=−3x+6y=23x−6
This expression shows that for any value of xxx, there is a corresponding value of yyy that satisfies the equation. The solution set is thus infinite, and can be written as: (x,y)=(x,3x−62)(x, y) = \left(x, \frac{3x – 6}{2}\right)(x,y)=(x,23x−6)
Hence, the correct answer is:
C. Infinitely many solutions
