What is the solution to the system? 2x +y =1 and x -y =2
To find the solution to the system of equations, we can use the method of substitution or elimination.
Using the method of substitution:
First, solve one of the equations for one variable in terms of the other. Let's solve the second equation for x:
x - y = 2
x = y + 2
Now substitute this expression for x into the first equation:
2(y + 2) + y = 1
2y + 4 + y = 1
3y + 4 = 1
3y = -3
y = -1
Substitute the value of y back into the second equation to find x:
x - (-1) = 2
x + 1 = 2
x = 2 - 1
x = 1
The solution to the system of equations is x = 1 and y = -1.
Using the method of elimination:
Multiply the second equation by 2 to eliminate y:
2(x - y) = 2(2)
2x - 2y = 4
Now add this equation to the first equation:
2x + y + 2x - 2y = 1 + 4
4x - y = 5
Rearrange the equation to solve for y:
-y = 5 - 4x
y = -5 + 4x
Now substitute this expression for y into either equation (let's use the first equation):
2x + (-5 + 4x) = 1
2x - 5 + 4x = 1
6x - 5 = 1
6x = 6
x = 1
Substitute the value of x back into the expression for y:
y = -5 + 4(1)
y = -5 + 4
y = -1
Again, the solution to the system of equations is x = 1 and y = -1.