Question

Samuel and Hayden solved the system of equations –6x – 6y = –6 and 7x + 7y = 7. Samuel says there are infinitely many solutions, but Hayden says the solution is (1,1). Explain who is correct.

Answer 1

Sample Response: The first step in solving this system is to multiply the first equation by 7 and the second equation by 6. When you add the equations, both variables are eliminated and you are left with 0 = 0, which is a true statement. That means there are infinitely many solutions and Samuel is correct.

Explanation:

Edge 2023

Answer 2

• Samuel is correct (infinitely many solutions)

Explanation:

You want to know if the system of equations has infinitely many solutions, or if there is one solution: (1, 1).

• -6x -6y = -6
• 7x +7y = 7

Standard form

We can put each of these equations in standard form by dividing both sides by the x-coefficient:

-6x -6y = -6 ⇒ x + y = 1

7x +7y = 7 ⇒ x + y = 1

We notice that these are the same equation. That means any (x, y) ordered pair that satisfies the first equation will also satisfy the second equation. There are infinitely many such ordered pairs, hence infinitely many solutions. Samuel is correct.

Hayden's solution

The proposed solution (1, 1) can be tried in these equations to see if it works:

1 + 1 = 1 . . . . . . false

(1, 1) is not a solution.