Given the following system is consistent
x+y=1
tx+y=t
(1+t)x+2y=3
Find the exact value of t.
subtract the first from the second
xt - x = t-1
x(t-1) = t-1
x = 1, if t±1
in 1st:
1+y = 1
y = 0
in the last:
(1+t)(1) + 0 = 3
1+t = 3
t = 2
Subtracting the first equation from the second gives you:
t-1 = (t-1)x
Therefore x = 1.
The first equation tells you that y = 0.
The third equation requires that
1 + t = 3
t = 2
To find the exact value of t, we will use the given system of equations and solve it step by step.
First, let's write down the given system of equations:
1) x + y = 1
2) tx + y = t
3) (1 + t)x + 2y = 3
We can solve this system using the method of elimination.
Let's begin by multiplying equation 1) by t:
t(x + y) = t(1)
tx + ty = t
Now, let's subtract equation 2) from the equation above:
tx + ty - tx - y = t - t
ty - y = 0
Factor out y:
y(t - 1) = 0
This equation tells us that either y = 0 or t - 1 = 0.
Case 1: y = 0
If y = 0, we can substitute y = 0 into equation 1):
x + 0 = 1
x = 1
Now, substitute x = 1 and y = 0 into equation 3):
(1 + t)(1) + 2(0) = 3
1 + t = 3
t = 3 - 1
t = 2
So, one solution is x = 1, y = 0, and t = 2.
Case 2: t - 1 = 0
If t - 1 = 0, then t = 1. Let's substitute t = 1 into equation 2):
(1)(x) + y = 1
x + y = 1
We can see that this is the same as equation 1).
Next, substitute x + y = 1 into equation 3):
(1 + 1)x + 2y = 3
2x + 2y = 3
Divide both sides by 2:
x + y = 3/2
We can see that this is not consistent with equation 1), so t = 1 does not satisfy the system.
Therefore, the only exact value of t that makes the system consistent is t = 2.
To find the exact value of t, we can use the given system of equations and solve for the variables x and y. Once we find x and y, we can substitute their values into the second equation to solve for t. Let's go step by step.
Step 1: Solve the first equation for x.
x + y = 1
We can rewrite this equation as:
x = 1 - y
Step 2: Substitute the value of x in the third equation.
(1+t)x + 2y = 3
Substituting 1 - y for x, we get:
(1+t)(1 - y) + 2y = 3
Simplifying this equation:
(1+t - ty) + 2y = 3
1 + t - ty + 2y = 3
1 + t + y(2 - t) = 3
t + y(2 - t) = 2
Step 3: Solve the second equation for y.
tx + y = t
Substituting 1 - y for x:
t(1 - y) + y = t
t - ty + y = t
y(1 - t) = 0
Step 4: Solve for y.
From the equation y(1 - t) = 0, we have two possibilities:
1 - t = 0 (Case 1)
This implies t = 1.
Or:
y = 0 (Case 2)
Step 5: Solve for t (Case 1).
In Case 1, where t = 1, we substitute the value of t into any of the original equations to solve for x and y.
Let's substitute t = 1 into the first equation:
x + y = 1
x + y = 1
Substituting y = 0 from Case 2:
x + 0 = 1
x = 1
In this case, we have the values x = 1, y = 0, and t = 1.
Therefore, the exact value of t is 1.