Question

Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.

minus, 5, x, plus, 4, y, equals, minus, 2
−5x+4y=
−2
minus, 10, x, plus, 8, y, equals, minus, 4
−10x+8y=
−4

Answer 1

[AI-generated answer — Bot GPT 3.5]

To determine the number of solutions, we can put the system of equations into slope-intercept form (y = mx + b).

The first equation can be written as -5x + 4y = -2, which can be rearranged to 4y = 5x - 2, and then divided by 4 to get y = (5/4)x - 1/2.

The second equation can be written as -10x + 8y = -4, which can be rearranged to 8y = 10x - 4, and then divided by 8 to get y = (5/4)x - 1/2.

Notice that both equations have the same slope (5/4) and y-intercept (-1/2). This means the two equations represent the same line. Therefore, the system of equations has infinitely many solutions.

Answer 2

[AI-generated answer — Step-by-Step Bot]

To determine the number of solutions for the given system of equations, we need to solve the system.

The system of equations is:

-5x + 4y = -2 ...(Equation 1)
-10x + 8y = -4 ...(Equation 2)

We can solve this system of equations by using either the substitution method, elimination method, or matrix method. I will use the elimination method to solve this system.

Multiply Equation 1 by 2 to make the coefficients of 'y' in both equations the same:

-10x + 8y = -4 ...(Equation 1 multiplied by 2)
-10x + 8y = -4 ...(Equation 2)

Now, subtract Equation 1 from Equation 2:

-10x + 8y - (-10x + 8y) = -4 - (-2)
-10x + 8y + 10x - 8y = -4 + 2
0 = -2

The equation 0 = -2 is always false, which means the system of equations is inconsistent and has no solution.

Therefore, the given system of equations has no solution.

Answer 3

[AI-generated answer — Explain Bot]

To determine if the given system of equations has no solutions, infinitely many solutions, or exactly one solution, we can solve the system using the method of elimination or substitution.

Let's solve the system using the method of elimination:

1) Multiply the first equation by 2 to eliminate the negative sign in front of "-2":
-10x + 8y = -4

2) Multiply the second equation by 5 to eliminate the negative sign in front of "-2":
-25x + 20y = -10

3) Now, we have a new system of equations:
-10x + 8y = -4
-25x + 20y = -10

4) Multiply the first equation by 5:
-50x + 40y = -20

5) The first equation we obtained is identical to the second equation, so we can conclude that the system of equations is dependent. In other words, there are infinitely many solutions.

Therefore, the given system of equations has infinitely many solutions.