Final answer:
The pair of equations that would have infinite solutions is option D. For two equations to have infinite solutions, they must be equivalent or represent the same line. After simplifying, option D's equations effectively become the same, confirming they have infinite solutions.
Step-by-step explanation:
The student has asked which pair of equations would have infinite solutions. To have infinite solutions, the two equations must represent the same line, which means they are identical when simplified or rearranged. Let's explore each option:
- A) y - 8 = 4z - 7, y = 4z - 7: Upon simplifying the first equation, we add 8 to both sides to yield y = 4z + 1, which does not match the second equation (y = 4z - 7). Thus, they are not identical.
- B) y = 7 - 4, y = -2: Simplifying the first equation gives y = 3, which is not equivalent to y = -2. Therefore, they do not have infinite solutions as they represent different lines.
- C) -3y = -3, z - 3y = 6: Dividing the first equation by -3 yields y = 1. Rearranging the second equation by subtracting z on both sides gives -3y = 6 - z. If we substitute y = 1 into the second equation, it simplifies to -3 = 6 - z, which is not possible for all values of z. Thus, they are not the same equation.
- D) 32 + y = 3, 6z + 2y = 6: Simplifying the first equation by subtracting 32 from both sides gives y = -29. Multiplying the first equation by 2 and then subtracting it from the second equation gives a redundant statement, 0 = 0, which means every solution of one equation is also a solution of the other. Therefore, they represent the same line and have an infinite number of solutions.
The pair of equations that would have infinite solutions is option D) 32 + y = 3, 6z + 2y = 6.