Question

y = x + 2
y = x + 1
click to select points on the graph.

Explanation:

Step1: Analyze the equations

The two equations are \( y = x + 2 \) and \( y = x + 1 \). These are linear equations in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. For \( y = x + 2 \), the slope \( m = 1 \) and the y - intercept \( b = 2 \). For \( y = x + 1 \), the slope \( m = 1 \) and the y - intercept \( b = 1 \).

Step2: Determine the relationship between the lines

Since both lines have the same slope (\( m = 1 \)) and different y - intercepts (\( 2
eq1 \)), the two lines are parallel. Parallel lines never intersect, so there is no solution for the system of equations formed by these two lines.

Step3: Graphing the lines (optional for understanding)

To graph \( y = x + 2 \), we can start at the y - intercept \( (0,2) \) and use the slope (rise 1, run 1) to find other points. For example, from \( (0,2) \), moving 1 unit up and 1 unit to the right gives \( (1,3) \), 1 unit up and 1 unit to the right from \( (1,3) \) gives \( (2,4) \), etc. For \( y = x + 1 \), start at the y - intercept \( (0,1) \), and using the slope (rise 1, run 1), we get points like \( (1,2) \), \( (2,3) \), etc. When we plot these points, we can see that the lines are parallel and do not meet.

Answer:

The system of equations \(

$$\begin{cases}y = x + 2\\y = x + 1\end{cases}$$

\) has no solution because the two lines are parallel (same slope, different y - intercepts).