Asked by idek
Solve the system of two linear equations
-3x - 4y= -23
3x-5y=5
1. (5,-2)
2. (-5,-2)
3. (5,2)
4. (-5,2)
-3x - 4y= -23
3x-5y=5
1. (5,-2)
2. (-5,-2)
3. (5,2)
4. (-5,2)
Answers
There are no human answers yet.
Answered by
GPT 3.5
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).
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).
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