Question

dose y=5x+7 and 3y-15x=18 have one solution, no solution or infinite solutions.

Answer 1

[AI-generated answer — Bot GPT 3.5]

To determine the solution(s) to the system of equations, we can solve for x and y.

First, we'll solve the first equation, y = 5x + 7, for y. By subtracting 5x from both sides, we get y = 5x + 7.

Now we'll substitute this expression for y into the second equation: 3y - 15x = 18.
Substituting y = 5x + 7, we get 3(5x + 7) - 15x = 18.
Expanding and simplifying this equation, we have 15x + 21 - 15x = 18, which simplifies to 21 = 18.

As 21 is not equal to 18, this equation is not true. This implies that there is no solution to the system of equations. Therefore, the system has no solution.

Answer 2

ok, what about x-2y=6 and 3x-6y=18

Answer 3

[AI-generated answer — Bot GPT 3.5]

To determine the solution(s) to the system of equations, we can solve for x and y.

We'll start by multiplying the first equation, x - 2y = 6, by 3 to get rid of the x coefficient in the second equation:
3(x - 2y) = 3(6)
Expanding this equation, we have 3x - 6y = 18.

Thus, we now have the system of equations:
3x - 6y = 18
3x - 6y = 18

Notice that the two equations are identical. This means that they represent the same line in the coordinate plane.

Since the two equations represent the same line, any point on this line satisfies both equations. Thus, there are infinite solutions to the system of equations.

In summary, the system has infinite solutions.

Answer 4

and what about y=2/3x-1 and y=2/3x - 2

Answer 5

[AI-generated answer — Bot GPT 3.5]

To determine the solution(s) to the system of equations, we can compare the two given equations:

y = (2/3)x - 1
y = (2/3)x - 2

Since the slope (2/3) and the y-intercept (-1 or -2) are the same for both equations, these two equations represent the same line. This means that any point on this line satisfies both equations.

Therefore, there are infinite solutions to the system of equations.

In summary, the system has infinite solutions.

Answer 6

and what about 3x+6=y and y=-1/3x-4

Answer 7

[AI-generated answer — Bot GPT 3.5]

To determine the solution(s) to the system of equations, we can compare the two given equations:

3x + 6 = y
y = (-1/3)x - 4

To find the solution, we can set the right-hand sides of the equations equal to each other:

3x + 6 = (-1/3)x - 4

To simplify this equation, we can multiply every term by 3 to eliminate the fractions:

9x + 18 = -x - 12

Combining like terms, we have:

10x = -30

Dividing both sides of the equation by 10, we find:

x = -3

Now we can substitute this value of x into one of the original equations to solve for y. Let's use the first equation:

3x + 6 = y
3(-3) + 6 = y
-9 + 6 = y
y = -3

Therefore, the solution to the system of equations is x = -3 and y = -3.

In conclusion, the system has one solution: (x = -3, y = -3).