Which term(s) describe(s) this system?
4x + y = 8
3x – 2y = –5
dependent
consistent and dependent
consistent and independent
inconsistent
consistent and independent because the answer is (1,4)
To determine which term(s) describe(s) the given system of equations, we need to solve the system and analyze the results.
Step 1: Rearrange both equations in the standard form (Ax + By = C):
4x + y = 8 => 4x + y - 8 = 0
3x - 2y = -5 => 3x - 2y + 5 = 0
Step 2: Write down the augmented matrix for the system:
[4 1 | 8]
[3 -2 | -5]
Step 3: Use the row reduction method (Gaussian elimination) to solve the system and obtain its row-echelon form:
[1 0 | -3]
[0 1 | 4]
Step 4: Analyze the solution:
The row-echelon form shows that the system has a unique solution, which means that the equations intersect at a single point. In other words, the system is consistent.
Since the system has a unique solution, it also implies that the two equations are not multiples of each other, meaning they are independent.
Therefore, the term(s) that describe(s) this system is "consistent and independent".