Pulling my hair, need help!

For the systems of linear equations in questions the following questions:

Determine how many solutions exist

Use either elimination or substitution to find the solutions (if any)

4x = 8 and 5y = 15

0.2x + 0.4y = 1.7 and 8.3x - 6.3y = -4.3

x/3 - y/5 = 4 and 3x/4 + 2y/3 = 6

4x=8 and 5y=15

x=2 y=3
02.x+0.4y=1.7 and 8.3x-6.3y=-4.3
x=1.93 and y=1.31

To determine how many solutions exist for each system of linear equations, we need to analyze their coefficients. Let's go through each system one by one:

1. For the system 4x = 8 and 5y = 15:
- In the first equation, we have only one variable, x. It is straightforward to solve for x by dividing both sides of the equation by 4. The result is x = 2.
- Similarly, in the second equation, we have only one variable, y. Dividing both sides of the equation by 5 gives us y = 3.
- Therefore, both equations have unique solutions, x = 2 and y = 3. This system has a unique solution.

2. For the system 0.2x + 0.4y = 1.7 and 8.3x - 6.3y = -4.3:
- Let's start by eliminating one variable. To do that, we can multiply the first equation by 10 to remove the decimal: 2x + 4y = 17.
- Now, we can attempt to eliminate the y variable by multiplying the second equation by 4: 33.2x - 25.2y = -17.2.
- By subtracting the first equation from the second equation, we get 31.2x - 29.2y = -34.2.
- This new equation does not simplify further, so we have two variables (x and y) and one equation. Therefore, we cannot determine a unique solution for this system. It has infinitely many solutions or no solution.

3. For the system x/3 - y/5 = 4 and 3x/4 + 2y/3 = 6:
- To eliminate the denominators, we can multiply the first equation by 15, resulting in 5x - 3y = 60.
- Similarly, multiplying the second equation by 12 gives us 9x + 8y = 36.
- By multiplying the first equation by 8 and the second equation by 3, we can create a system where we can eliminate one variable: 40x - 24y = 480 and 27x + 24y = 108.
- By adding the equations together, we get 67x = 588, and dividing both sides by 67 gives us x ≈ 8.776.
- Substituting this value of x into either of the original equations, we can solve for y. Using the first equation, y = (15x - 60)/3, we find y ≈ 24.706.
- Therefore, this system has a unique solution, x ≈ 8.776 and y ≈ 24.706.

In summary:
- The first system has a unique solution: x = 2 and y = 3.
- The second system has either infinitely many solutions or no solution.
- The third system has a unique solution: x ≈ 8.776 and y ≈ 24.706.