Solve the system of two linear equations

Question

-3x - 4y= -23
3x-5y=5

1. (5,-2)
2. (-5,-2)
3. (5,2)
4. (-5,2)

Answer 1

To solve the system of equations, we can use the method of elimination.

First, let's multiply the second equation by 4 to get:

12x - 20y = 20

Now we can subtract this equation from the first equation:

(-3x - 4y) - (12x - 20y) = (-23) - (20)
-3x - 4y - 12x + 20y = -43
-15x + 16y = -43

Now we have a new equation:
-15x + 16y = -43

Next, we can add this equation to the original second equation:

(-15x + 16y) + (3x - 5y) = (-43) + (5)
-15x + 16y + 3x - 5y = -38
-12x + 11y = -38

Now we have another new equation:
-12x + 11y = -38

We now have a system of two equations:
-15x + 16y = -43
-12x + 11y = -38

To solve this system by elimination, we can multiply the second equation by -16 and the first equation by 11:

-15*(-16)x + 16*(-16)y = -43*(-16)
-12*(-11)x + 11*(-11)y = -38*(-11)

240x - 256y = 688
132x - 121y = 418

Now we can add these two equations together:

240x - 256y + 132x - 121y = 688 + 418
372x - 377y = 1106

Now we have a new equation:
372x - 377y = 1106

This equation can be rewritten as:
372x = 377y + 1106

To find integer solutions for x and y, we need to find a common multiple of 372 and 377 that leaves a remainder of 1106 when divided by 377.

By trying different multiples of 377, we find that the smallest common multiple of 372 and 377 that leaves a remainder of 1106 when divided by 377 is 5,607,129.

Therefore, we have:
x = (5,607,129 + 377y)/372

To find integer solutions, y must be equal to -2. Plugging this into the equation for x:

x = (5,607,129 + 377*(-2))/372
x = 15,087/372
x = 5

So the solution to the system of equations is (5, -2).

Therefore, the correct answer is 1. (5, -2).