The given system of equations is:
1) x + 3y = 11
2) y = x - 7
To determine which statement is true about this system, we can solve the equations by substitution or elimination:
Using the substitution method:
1) Substitute the value of y from equation 2 into equation 1:
x + 3(x - 7) = 11
Simplify:
x + 3x - 21 = 11
Combine like terms:
4x - 21 = 11
Add 21 to both sides:
4x = 32
Divide by 4:
x = 8
Substitute x = 8 into equation 2 to find y:
y = 8 - 7
y = 1
Therefore, the solution to the system of equations is x = 8 and y = 1.
Now, let's analyze the statements:
Statement 1: The system has no solution.
This statement is false because we have found a unique solution for the system, which is x = 8 and y = 1.
Statement 2: The system has infinitely many solutions.
This statement is false because we have found a single solution for the system, not infinitely many.
Statement 3: The system has exactly one solution.
This statement is true because we have found one unique solution for the system, which satisfies both equations.
In summary, statement 3 is true: The system x + 3y = 11 and y = x - 7 has exactly one solution, which is x = 8 and y = 1