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. \\( y = x + 4 \\) \\( 2x - 2y = 2 \\) \\( \text{a) } 1 \text{ solution; consistent and independent} \\) \\( \text{b) infinitely many solutions; consistent and dependent} \\) \\( \text{c) } 1 \text{ solution; consistent and dependent} \\) \\( \text{d) no solution; inconsistent} \\)

Explanation:

Step1: Analyze the first equation

The first equation is \( y = x + 4 \). Its slope-intercept form is clear, with a slope of \( 1 \) and a y-intercept of \( 4 \).

Step2: Rewrite the second equation

Rewrite \( 2x - 2y = 2 \) in slope-intercept form. Divide both sides by \( 2 \): \( x - y = 1 \), then \( y = x - 1 \). This equation has a slope of \( 1 \) and a y-intercept of \( -1 \).

Step3: Compare slopes and intercepts

Both lines have the same slope (\( 1 \)) but different y-intercepts (\( 4 \) and \( -1 \)). Parallel lines (same slope, different intercepts) never intersect, so there are no solutions.

Step4: Determine consistency and dependence

A system with no solutions is inconsistent (since consistent systems have at least one solution). Independent/dependent applies to consistent systems; here, it's inconsistent.

Answer:

D) no solution; inconsistent