How many solutions are there to the following system of equations?
−8=−4y+x
−54x=y+4
4 y = x + 8
y = -54 x - 4
4 y = x + 8
4y = -216 x -16
so
x+8 = -216 x - 16
217 x = -24
x = -24/217
then y = -54 x - 4
Those lines only hit once :)
Write first equation as:
4 y = - x - 8
Divide both sides by 4
y = - x / 4 - 2 = - 1 / 4 x - 2
Write second equation as:
y = - 54 x - 4
The gradient of first line:
m1 = - 1 / 4
The gradient of second line:
m2 = - 54
Two straight line with different gradients have only one point of intersection.
So you system have one solution.
My typo.
Write first equation as:
4 y = x + 8
Divide both sides by 4
y = x / 4 + 2 = 1 / 4 x + 2
Write second equation as:
y = - 54 x - 4
The gradient of first line:
m1 = 1 / 4
The gradient of second line:
m2 = - 54
Two straight line with different gradients have only one point of intersection.
So you system have one solution.
To determine the number of solutions for the given system of equations, we can solve the equations simultaneously and then analyze the solution.
Let's solve the system of equations step by step:
1) Start with the given system of equations:
−8 = −4y + x ---(1)
−54x = y + 4 ---(2)
2) Rearrange equation (2) to solve for y:
−54x - 4 = y ---(3)
3) Substitute equation (3) into equation (1) to eliminate the variable y:
−8 = −4(−54x - 4) + x
−8 = 216x + 16 + x
−8 = 217x + 16
−8 -16 = 217x
−24 = 217x
4) Solve for x:
x = -24/217
5) Substitute the value of x back into equation (3) to solve for y:
y = −54(-24/217) - 4
y = 1296/217 - 868/217
y = 428/217
So, we have found the values of x and y to be x = -24/217 and y = 428/217.
Now, let's analyze the solution:
Since x and y have specific values, it means that the system of equations has one unique solution. There is only one point (x, y) where both equations are satisfied simultaneously.