How can we determine if the system of equations 2x + y = 9 and 3x + 4y = 8 is independent, dependent, or inconsistent without graphing?

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To determine if the system of equations is independent, dependent, or inconsistent, we can analyze the equations algebraically without graphing.

The two equations in the system are:

  • Equation 1: 2x + y = 9
  • Equation 2: 3x + 4y = 8

We can start by expressing one variable in terms of the other using Equation 1. Let’s solve for y:

y = 9 - 2x

Next, we can substitute this expression for y into Equation 2:

3x + 4(9 - 2x) = 8

Now simplify this equation:

3x + 36 - 8x = 8
-5x + 36 = 8
-5x = 8 - 36
-5x = -28
x = rac{28}{5}

Now that we have the value of x, we can substitute it back to find y:

y = 9 - 2(rac{28}{5})
y = 9 - rac{56}{5}
y = rac{45}{5} - rac{56}{5}
y = -rac{11}{5}

Thus, we have found a unique solution: x = rac{28}{5} and y = - rac{11}{5}. Since we obtained a single unique solution, the system of equations is classified as independent.

In conclusion, after solving the equations, we determined that the system is independent because there is a unique solution.

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