Compute x +y, given the linear system with T # 1 and with the ordered pair (x,y) as solution
x + 2y=3
Tx + ( T+1)y = T + 2
To compute the values of x and y, we need to solve the given linear system of equations.
Let's start by multiplying the first equation by T+1 and the second equation by -2:
(T+1)(x + 2y) = 3(T+1)
-2(Tx + (T+1)y) = -2(T+2)
Simplifying these equations, we get:
(T^2+T)x + (2T+2y) = 3T+3
-2Tx - 2(T+1)y = -2T-4
Now, let's eliminate the x term by adding the two equations:
(T^2+T)x - 2Tx + (2y - 2(T+1)y) = 3T+3 -2T-4
Simplifying further,
(T^2+T - 2T)x + (2y - 2Ty - 2y) = 3T+3 -2T-4
(T^2-T)x - 2Ty + 0 = T-1
Simplifying again,
(T^2-T)x - 2Ty = T-1
Now, since the ordered pair (x,y) is a solution to this system, the x and y values should satisfy the equation above.
Therefore, we have:
(T^2-T)x - 2Ty = T-1
Plug in T=1 into the equation above:
(1^2-1)x - 2y = 1-1
(0)x - 2y = 0
0 - 2y = 0
-2y = 0
Solving for y, we get:
y = 0
Now, substitute y=0 into the first equation of the original system:
x + 2(0) = 3
x + 0 = 3
x = 3
So, the solution to the system is x = 3 and y = 0.
To compute x + y, we simply add the values:
x + y = 3 + 0 = 3.