Question

use the graph to determine the number of solutions the system has. then state whether the system of equations is consistent or inconsistent and if it is independent or dependent. $4x + 6y = 12$ $2x + 3y = 6$ a) 1 solution; consistent and independent b) 1 solution; consistent and dependent c) no solution; inconsistent d) infinitely many solutions; consistent and dependent

Explanation:

Step1: Simplify the equations

First, simplify the two equations. For the equation \(4x + 6y = 12\), we can divide the entire equation by 2, which gives \(2x + 3y = 6\). Notice that this is the same as the second equation \(2x + 3y = 6\).

Step2: Analyze the system

Since the two equations are actually the same line (they represent the same linear equation), the system of equations has infinitely many solutions. A system with infinitely many solutions is consistent (because there are solutions) and dependent (because the equations are not independent; one is a multiple of the other).

Answer:

D) infinitely many solutions; consistent and dependent