A 3D object is represented by the system of equations: x + 2y + z = 5
The system has infinitely many solutions. Which of the following equations could describe another line/plane that is part of the same 3D object?

A) x + 2y + 2z = 7

B) 2x + 4y + 2z = 10

C) 3x + 6y + 3z = 12

D) x - y + z = 0

The given system is:
x + 2y = 4 (Equation 1)
mx + (2m - 1)y = n (Equation 2)
For what values of m and n will the system have infinitely many solutions?

A) m = 1, n = 2

B) m = any real number, n = a multiple of 4

C) m = any even integer, n = any real number

D) m = -2, n = 0

A moving train is represented by the equation:
y = mx + b (initial position and slope)
Another object is also moving along the same track. The system has infinitely many solutions. What can you determine about the second object’s movement?

A) The second object is moving on a perpendicular track.

B) The second object is always ahead of the train.

C) The second object is moving on a parallel track with the same constant relative speed.

D) The second object is moving on a parallel track with a different speed.

The system of equations is:
x + y = 5 (Equation 1)
2x + ky = 10 (Equation 2)
Express all solutions of the system in parametric form (using one variable dependent on another).

A) x = any real number, y = 5 - 2x

B) x = y + 2, y = any real number

C) x = t, y = 5 - t

D) x = 2t + 1, y = 3 - t

A rectangle has a fixed width of 4 units. The system of equations below describes two lines that are part of the rectangle’s boundaries.
y = 2x + b1 (Equation 1) - Represents one side
x + ay = a (Equation 2) - Represents another side (a positive constant)
Under what condition(s) will the system have infinitely many solutions? How will this be reflected geometrically in the rectangle?

A) a = 2 and b1 = any real number

B) a = any positive constant and b1 = 4

C) There are no conditions for infinite solutions.

D) a = \$1/2\$ and b1 = any real number.

In the system of equations below, a and c are constants.
\$ (1/2)x + (1/3)y = (1/6) \$
\$ ax + y = c\$
If the system of equations has an infinite number of solutions(x, y), what is the value of a?

A) \$ -1/2 \$

B) 0

C) \$ 3/2 \$

D) \$ 1/2 \$

A financial advisor suggests investing in two different portfolios. Portfolio A yields a 10% return annually, while Portfolio B yields a 15% return annually. If a total investment of $50,000 is made and the total return is $6,000 annually, determine the optimal distribution of investment between the two portfolios.

A) $35,000 in Portfolio A and $15,000 in Portfolio B

B) $30,000 in Portfolio A and $20,000 in Portfolio B

C) $10,000 in Portfolio A and $40,000 in Portfolio B

D) $20,000 in Portfolio A and $30,000 in Portfolio B

An artist is mixing two colors of paint to create a new color. One gallon of red paint costs $10, and one gallon of blue paint costs $15. The artist wants to create a mixture that costs $12 per gallon. How much of each color should be used to create a 10-gallon mixture?

A) 6 gallons of red paint and 4 gallons of blue paint

B) 5 gallons of red paint and 5 gallons of blue paint

C) 4 gallons of red paint and 6 gallons of blue paint

D) 7 gallons of red paint and 3 gallons of blue paint

In the system of equations above, u and v are constants. If the system has infinitely many solutions, what is the value of uv:
- 7x - 5y - 10 = 0 and - ux - vy - 100 = 0

A) -55

B) -35

C) -45

D) -25

Show that the following system of equations has the infinite solution: 5x + 8y = 10 and 25x + 40y = 50

A) \$ (a_1)/(a_2) ne (b_1)/(b_2) = (c_1)/(c_2) \$

B) \$ (a_1)/(a_2) ne (b_1)/(b_2) ne (c_1)/(c_2) \$

C) \$ (a_1)/(a_2) = (b_1)/(b_2) = (c_1)/(c_2) \$

D) \$ (a_1)/(a_2) = (b_1)/(b_2) ne (c_1)/(c_2) \$