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.

$y = 2x - 3$
$2x - 2y = 2$

a) 1 solution; consistent and independent
b) infinitely many solutions; consistent and dependent
c) 1 solution; consistent and dependent
d) no solution; inconsistent

Explanation:

Step1: Analyze the equations' slopes

First, rewrite \(2x - 2y = 2\) in slope - intercept form (\(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept).
Starting with \(2x-2y = 2\), subtract \(2x\) from both sides: \(-2y=-2x + 2\).
Divide both sides by \(-2\): \(y=x - 1\).
The equation \(y = 2x-3\) has a slope \(m_1=2\), and the equation \(y=x - 1\) (from \(2x - 2y=2\)) has a slope \(m_2 = 1\). Since \(m_1
eq m_2\), the two lines are not parallel, so they must intersect at exactly one point.

Step2: Determine consistency and dependence

A system of linear equations is consistent if it has at least one solution. Since the two lines intersect (one solution), the system is consistent.
A system of linear equations is independent if the two equations represent two distinct lines (not coinciding). Since the two lines have different slopes (and different equations), they are distinct lines, so the system is independent.

Step3: Number of solutions

Since the two lines intersect at one point, the system has 1 solution.

Answer:

A) 1 solution; consistent and independent