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.
$x - y = -4$
$y = x + 4$
a) 1 solution; consistent and independent
b) infinitely many solutions; consistent and dependent
c) 1 solution; consistent and dependent
d) no solution; inconsistent

Explanation:

1. First, rewrite the first equation \(x - y=-4\) in slope - intercept form (\(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept). Solve for \(y\):

• Start with \(x - y=-4\). Subtract \(x\) from both sides: \(-y=-x - 4\). Multiply both sides by \(-1\) to get \(y=x + 4\).

1. The second equation is already \(y=x + 4\). This means that the two equations represent the same line.

• For a system of linear equations \(y = m_1x + b_1\) and \(y=m_2x + b_2\), if \(m_1=m_2\) and \(b_1 = b_2\), the lines are coincident (they lie on top of each other).
• A system of linear equations with coincident lines has infinitely many solutions.
• A consistent system is one that has at least one solution. Since this system has infinitely many solutions, it is consistent. A dependent system is one where the equations are not independent (they represent the same line or are scalar multiples of each other), so this system is dependent.

Answer:

B) infinitely many solutions; consistent and dependent