To determine whether the system of equations is independent, dependent, or inconsistent, we can analyze the equations algebraically.
The given equations are:
- 1) 2x + y = 9
- 2) 3x + 4y = 8
First, we can express both equations in terms of one variable, say y.
From the first equation:
y = 9 – 2x
Now, substituting this expression for y into the second equation:
3x + 4(9 – 2x) = 8
This simplifies to:
- 3x + 36 – 8x = 8
- -5x + 36 = 8
- -5x = 8 – 36
- -5x = -28
- x = rac{28}{5}
Now that we have a value for x, we can substitute it back to find y:
y = 9 – 2(28/5)
y = 9 – 56/5
y = rac{45}{5} – rac{56}{5}
y = - rac{11}{5}
Since we found a unique solution for both x and y, this means that the system of equations is independent. An independent system has exactly one solution, whereas a dependent system has infinitely many solutions, and an inconsistent system has no solution.
In conclusion, the system of equations 2x + y = 9 and 3x + 4y = 8 is independent.