Question

Which of the following systems of equations has infinitely many solutions?

Answer 1

When two equations are equivalent, it is infinitely many solutions because the two lines overlap each other.

Option A - 2x + 7x + y = 6 and y = 9x + 6

• => (9x + y = 6) = (y = 9x + 6)
• => (y = -9x + 6) ≠ (y = 9x + 6)

Option B - y = 5x + 7 and y = 2x + 8

• => (y = 5x + 7) ≠ (y = 2x + 8)

Option C - y = x + 4 and 2y = 2x + 8

• => (y = x + 4) = (2y = 2x + 8)
• => (y = x + 4) = (y = 2x/2 + 8/2)
• => (y = x + 4) = (y = x + 4)

Option D - 7x - y = 10 and y = 6x + 8

• => (7x - y = 10) = (y = 6x + 8)
• => (-y = -7x + 10) = (y = 6x + 8)
• => (y = 7x - 10) ≠ (y = -6x - 8)

Hence, Option C is correct.

Answer 2

• C. y = x + 4, 2y = 2x + 8

Explanation:

The system has infinitely many solutions if the two equations are equivalent.

We can analyze the given systems to see:

A)

• 2x + 7x + y = 6 ⇒ 9x + y = 6 ⇒ y = - 9x + 6
• y = 9x + 6

Different slopes, there is one solution

B)

• y = 5x + 7
• y = 2x + 8

Different slopes, there is one solution

C)

• y = x + 4
• 2y = 2x + 8 ⇒ y = x + 4

Same lines, infinitely many solutions

D)

• 7x - y = 10 ⇒ y = 7x - 10
• y = 6x + 8

Different slopes, there is one solution