Question

which graph represents this system?
y = 3
x + y = 4

Explanation:

Step1: Analyze \( y = 3 \)

The equation \( y = 3 \) is a horizontal line. It has a \( y \)-intercept at \( (0, 3) \) and is parallel to the \( x \)-axis, meaning for all values of \( x \), \( y \) remains 3.

Step2: Analyze \( x + y = 4 \)

Rewrite \( x + y = 4 \) in slope - intercept form (\( y=mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept). Solve for \( y \): \( y=-x + 4 \). The slope \( m=-1 \) and the \( y \)-intercept \( b = 4 \) (so the line passes through \( (0,4) \)). To find the \( x \)-intercept, set \( y = 0 \): \( 0=-x + 4\Rightarrow x = 4 \), so the line also passes through \( (4,0) \).

Step3: Find the intersection point

We have the system \(

$$\begin{cases}y = 3\\y=-x + 4\end{cases}$$

\). Substitute \( y = 3 \) into \( y=-x + 4 \): \( 3=-x + 4\Rightarrow x=4 - 3=1 \). So the two lines intersect at \( (1,3) \).

To identify the graph: One line is horizontal at \( y = 3 \), the other has a slope of - 1, \( y \)-intercept at \( (0,4) \) and \( x \)-intercept at \( (4,0) \), and they intersect at \( (1,3) \).

Answer:

The graph should consist of a horizontal line \( y = 3 \) and a line with slope - 1 passing through \( (0,4) \) and \( (4,0) \) that intersect at \( (1,3) \). (Since the actual graph options are not fully provided here, but the key features to look for are as described above)