Let’s address the questions in the image step by step:

Items 1-4:

The scenario describes a dog that starts at 1 kg at birth and gains 5 kg per month. Let's answer each question:

1. Predict the dog's weight after 5 months using a mathematical equation and graphical representation.

   • The weight $W$ can be modeled using the linear equation: $W = 5m + 1$ where $m$ represents the number of months.
   • After 5 months: $W = 5(5) + 1 = 26 \, \text{kg}.$

2. What is the dependent variable?

   • The dependent variable is the dog's weight $W$, as it depends on the number of months.

3. What is the equation of the line?

   • The equation is: $W = 5m + 1$

4. What is the value of the slope?

   • The slope is 5, indicating that the dog gains 5 kg per month.

Item 5:

Consider the system of equations:

$2x + 3y = 6 \quad \text{and} \quad 4x + 6y = 12$

• Notice that the second equation is a multiple of the first: $4x + 6y = 2(2x + 3y).$
• This means the equations are dependent, and the system has infinitely many solutions (the two lines are the same).

Item 6:

Graph of the system of equations:

$y = 2x + 4 \quad \text{and} \quad y = 2x - 2$

• Both lines have the same slope ($m = 2$) but different $y$-intercepts ($4$ and $-2$).
• Since they are parallel, the system has no solution.

The correct graph is D.

Item 7:

Graph of the system of equations:

$y = -3x + 4 \quad \text{and} \quad y = x - 2$

• These two lines have different slopes ($-3$ and $1$), so they intersect at a single point.

The correct graph is B.

Item 8:

Given $4y - 3x = 9$, find the slope in slope-intercept form:

1. Rewrite the equation into slope-intercept form ($y = mx + b$): $4y = 3x + 9 \quad \implies \quad y = \frac{3}{4}x + \frac{9}{4}.$
2. The slope ($m$) is: $m = \frac{3}{4}.$

Item 9:

Given $6y - 2x = 18$, find the $y$-intercept:

1. Rewrite the equation into slope-intercept form ($y = mx + b$): $6y = 2x + 18 \quad \implies \quad y = \frac{1}{3}x + 3.$
2. The $y$-intercept ($b$) is: $b = 3.$

Would you like further clarifications on any of these solutions or a graphical explanation?

Follow-Up Questions:

1. Can you describe how to determine if a system of equations is dependent, independent, or inconsistent?
2. How do parallel lines in a graph relate to the slope of the equations?
3. What does the slope of a line represent in a real-world context like the dog’s weight?
4. How can you graphically determine the solution to a system of equations?
5. How would the equation change if the dog gained weight at a decreasing rate?

Tip:

To easily convert standard form equations to slope-intercept form, isolate $y$ step by step to uncover the slope and intercept!