Question

is there an ordered pair that is a solution for this system of linear equations? how many solutions are there?
*you can type them into the desmos calculator to see the graph.

$y = x + 1$
$y = x + 5$

\bigcirc\\ one solution

\bigcirc\\ no solution

\bigcirc\\ infinitely many solutions

Explanation:

Step1: Analyze slope and intercept

The two equations are in slope - intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. For \(y=x + 1\), the slope \(m_1=1\) and y - intercept \(b_1 = 1\). For \(y=x + 5\), the slope \(m_2 = 1\) and y - intercept \(b_2=5\).

Step2: Determine the relationship between lines

Since the slopes of the two lines are equal (\(m_1=m_2 = 1\)) and the y - intercepts are different (\(b_1
eq b_2\)), the two lines are parallel. Parallel lines in a plane never intersect.

Step3: Relate intersection to solutions

The solution of a system of linear equations is the point of intersection of their graphs. If two lines are parallel and do not intersect, there is no ordered pair \((x,y)\) that satisfies both equations simultaneously.

Answer:

No solution