Question

which system of equations best represents the system shown on the graph?
a ( y = x + 2 )
( y = - 4x + 17 )

Explanation:

Step1: Analyze the first line \( y = x + 2 \)

The slope - intercept form of a line is \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. For the line \( y=x + 2 \), the slope \( m = 1\) (since the coefficient of \( x \) is 1) and the y - intercept \( b = 2\). This means the line crosses the y - axis at \( (0,2) \) and has a positive slope (it rises from left to right).

Step2: Analyze the second line \( y=-4x + 17 \)

For the line \( y=-4x + 17 \), the slope \( m=-4\) (negative slope, so it falls from left to right) and the y - intercept \( b = 17\).

Step3: Check the intersection and the graph

We can also find the intersection point of the two lines \( y=x + 2 \) and \( y=-4x + 17 \) by setting them equal to each other:
\( x + 2=-4x+17 \)
Add \( 4x \) to both sides: \( x + 4x+2=-4x + 4x+17\)
\( 5x+2 = 17 \)
Subtract 2 from both sides: \( 5x+2 - 2=17 - 2\)
\( 5x=15 \)
Divide both sides by 5: \( x = 3 \)
Substitute \( x = 3 \) into \( y=x + 2 \), we get \( y=3 + 2=5 \). So the intersection point is \( (3,5) \).

Looking at the graph, we can see that one line has a positive slope (consistent with \( y=x + 2 \)) and one line has a negative slope (consistent with \( y=-4x + 17 \)). The y - intercepts and the slopes match the visual representation of the lines in the graph.

Answer:

\(

$$\begin{cases}y = x+2\\y=-4x + 17\end{cases}$$

\) (the system with equations \( y = x + 2 \) and \( y=-4x + 17 \))