https://file.aiquickdraw.com/mathbot/file/17268865494502ujvb7h5.jpg

Let's analyze each system of equations presented in the image and predict the number of solutions.

### a) \( y = 45,000 + 200x \) and \( y = 35,000 + 300x \)

Here, we are comparing two linear equations. The general form of a linear equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

- In the first equation, \( m = 200 \) and \( b = 45,000 \).
- In the second equation, \( m = 300 \) and \( b = 35,000 \).

Since the slopes (\( m \)) are different, the lines are not parallel. Therefore, they intersect at exactly one point.

**Correct answer:** *One solution* because the lines have *different slopes*.

### b) \( y = 45,000 + 200x \) and \( y = 35,000 + 200x \)

In this case:
- Both equations have the same slope: \( m = 200 \).
- The y-intercepts are different: \( b_1 = 45,000 \) and \( b_2 = 35,000 \).

When lines have the same slope but different y-intercepts, they are parallel and will never intersect.

**Correct answer:** *No solution* because the lines have *the same slope but different y-intercepts*.

### c) \( y = 45,000 + 200x \) and \( y = 45,000 + 300x \)

- In the first equation, \( m = 200 \) and \( b = 45,000 \).
- In the second equation, \( m = 300 \) and \( b = 45,000 \).

Although the y-intercepts are the same, the slopes are different, meaning the lines will intersect at exactly one point.

**Correct answer:** *One solution* because the lines have *different slopes*.

Let me know if you have further questions or would like additional explanations! Here are five related questions for further understanding:

1. What happens if two lines have the same slope and the same y-intercept?
2. How do you find the point of intersection for two lines?
3. What conditions must be met for a system of equations to have infinitely many solutions?
4. Can two lines with different slopes ever be parallel?
5. How can you tell from an equation whether two lines are perpendicular?

**Tip:** When determining the number of solutions in a system of linear equations, always compare the slopes first. Different slopes mean one solution (the lines intersect). The same slope with different intercepts means no solution (the lines are parallel).

Predict the number of solutions to each of the following systems. Give reasons for your answer.

Analyze systems of linear equations by comparing their slopes and y-intercepts. Learn to determine whether there are one, no, or infinitely many solutions. Suitable for students in Grades 9-10.

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