Question

How many solutions does this equation have

12x-15y=18
4x-5y=6

Answer 1

[AI-generated answer — Bot GPT 3.5]

To determine the number of solutions, we need to examine their relationship.

Let's multiply the first equation by 4 and the second equation by 3 to make the coefficients of x equal.

The new equations become:

48x - 60y = 72
12x - 15y = 18

To simplify, we can divide all the terms by 3 in the first equation:

16x - 20y = 24

Now, we can see that both equations are the same. They represent the same line.

Since the two equations represent the same line, there are infinitely many solutions.

Answer 2

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

To find the number of solutions for the system of equations, we can use the concept of determinants.

The given system of equations can be written in matrix form as:

| 12 -15 | | x | = | 18 |
| 4 -5 | | y | | 6 |

To determine the number of solutions, we need to calculate the determinant of the coefficient matrix:

D = | 12 -15 |
| 4 -5 |

Using the formula for a 2x2 matrix, the determinant D is calculated as:

D = (12 * -5) - (-15 * 4)
D = -60 + 60
D = 0

If the determinant (D) is equal to zero, then the system has infinitely many solutions. If the determinant is non-zero, the system has a unique solution.

In this case, since the determinant is 0, the system has infinitely many solutions.

Answer 3

[AI-generated answer — Explain Bot]

To determine the number of solutions for this system of equations, we can use the concept of linear dependence. If the two equations are linearly dependent, it means they represent the same line, and there are infinitely many solutions. If the equations are linearly independent, it means they represent distinct lines that intersect at a single point, and there is one unique solution.

Let's manipulate the equations to determine their linear dependence.

Equation 1: 12x - 15y = 18
Equation 2: 4x - 5y = 6

To check linear dependence, we can divide the coefficients of one equation by the coefficients of the other equation and observe if the ratios are the same.

Let's divide Equation 1 by Equation 2:
(12x - 15y) / (4x - 5y) = (18) / (6)
(12x - 15y) / (4x - 5y) = 3

Further simplifying, we can distribute 3 to the denominator:
(12x - 15y) / [(4x - 5y) * 1] = 3

Now, let's equate the numerator of Equation 1 to the numerator of the ratio:
12x - 15y = 3

We can see that Equation 1 is a simple simplification of Equation 2, suggesting that they represent the same line. Therefore, their system of equations is linearly dependent.

As a result, the system of equations has infinitely many solutions.