Let's go through the questions step by step.

Question 4

The system of equations provided is:

1. $2y + x = 4$
2. $y = 3x + 9$

Solution:
To graph the system:

• Rewrite the first equation $2y + x = 4$ into slope-intercept form:
  $2y = -x + 4 \quad \Rightarrow \quad y = -\frac{1}{2}x + 2$ This line has a slope of $-\frac{1}{2}$ and a $y$-intercept of $2$.
• The second equation is already in slope-intercept form:
  $y = 3x + 9$ This line has a slope of $3$ and a $y$-intercept of $9$.

Graph these two lines:
The solution to the system is the point where these two lines intersect. Based on their slopes, they will intersect at exactly one point. Without graphing explicitly, solve algebraically:
Set $-\frac{1}{2}x + 2 = 3x + 9$:
$-\frac{1}{2}x - 3x = 9 - 2$ Combine like terms:
$-\frac{7}{2}x = 7$ Solve for $x$:
$x = -2$ Substitute $x = -2$ into either equation (e.g., $y = 3x + 9$):
$y = 3(-2) + 9 = -6 + 9 = 3$ Solution: The lines intersect at $(-2, 3)$.

Question 5

Which statements describe the solution to a linear system of equations?

Correct answers:

• A: The point(s) of intersection of the graphs of each equation.
• E: A coordinate pair that makes both equations true.
• F: A point on the graph of both lines.

Question 6

The given system is:

1. $3x - y = 2$
2. $6x - 2y = 4$

Solution:
Notice that the second equation is a multiple of the first:
$6x - 2y = 2(3x - y) = 4$ This means the two equations represent the same line. Therefore, the system has infinitely many solutions because every point on the line satisfies both equations.

Question 7

Analyzing the graph:

• Aniyah says the system has no solution because the lines do not cross and are growing apart.
• Jessica says the system has a solution because the lines are not parallel.

From the graph, the lines are parallel because they have the same slope but different $y$-intercepts. Parallel lines never intersect, so there is no solution.

Correct answer: Aniyah is correct because the lines are parallel.

Summary Answers:

1. Question 4: Solution is $(-2, 3)$.
2. Question 5: A, E, F are correct.
3. Question 6: Infinitely many solutions (same line).
4. Question 7: Aniyah is correct (no solution; lines are parallel).

Follow-up Questions:

1. How do you identify parallel lines using slopes?
2. What are the different possible solutions for systems of linear equations?
3. How can you graph a linear equation given its slope and $y$-intercept?
4. Why do two identical equations represent infinitely many solutions?
5. How do you solve a system of equations algebraically?

Tip:

To check if two lines are parallel, compare their slopes. If they are equal but the $y$-intercepts differ, the lines are parallel and have no solution.