Question

Match the system of equations to the number of solutions it has.

You can use your Graphing Calculator!

(4 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
y = 5x + 7
_____________
3y - 15x = 18

y = 3x + 6
________________
y = -13x - 4

x - 2y = 6
_____________________
3x - 6y = 18

y = 23x - 1
_____________________
y = 23x - 2

Options: One solution Or No solution Or Infinite solutions

Answer 1

[AI-generated answer — Bot GPT 3.5]

- y = 5x + 7

_____________
3y - 15x = 18

- y = 3x + 6
________________
y = -13x - 4

- x - 2y = 6
_____________________
3x - 6y = 18

- y = 23x - 1
_____________________
y = 23x - 2

- Options:

- One solution
- No solution
- Infinite solutions

Answer 2

[AI-generated answer — Step-by-Step Bot]

To determine the number of solutions for each system of equations, we can compare their slopes. If the slopes are equal, the lines are parallel and there is no solution. If the slopes are different, the lines intersect at a single point and there is one solution. If the slopes are equal and the y-intercepts are also equal, the lines are coincident and there are infinite solutions.

Let's analyze the given systems one by one:

1. y = 5x + 7
3y - 15x = 18

Both equations are in slope-intercept form (y = mx + b), where the slopes (m) are 5 and -5, respectively. Since the slopes are not equal, these lines intersect at a single point and there is one solution.

Answer: One solution

2. y = 3x + 6
y = -13x - 4

Again, both equations are in slope-intercept form, but this time the slopes are 3 and -13, respectively. Once again, the slopes are not equal, so there is one solution.

Answer: One solution

3. x - 2y = 6
3x - 6y = 18

Now, these equations are in standard form, which we'll convert to slope-intercept form first. The first equation can be rearranged to y = 0.5x - 3, and the second equation can be simplified to x - 2y = 6. The slopes of both lines are equal (0.5), but the y-intercepts are different. Therefore, the lines are parallel and there is no solution.

Answer: No solution

4. y = 23x - 1
y = 23x - 2

These equations are in slope-intercept form and both have the same slope (23). However, the y-intercepts are different, indicating that the lines are parallel and there is no solution.

Answer: No solution

Summary:
1. One solution
2. One solution
3. No solution
4. No solution

Answer 3

[AI-generated answer — Explain Bot]

To determine the number of solutions for each system of equations, you can use your graphing calculator. Here's how:

1. Enter the first equation, y = 5x + 7, into the equation editor of your graphing calculator.
2. Enter the second equation, 3y - 15x = 18, into the equation editor as well.
3. Graph both equations on the same coordinate system using the graphing function of your calculator.
4. Look at the intersection point(s) of the two graphs.

Now let's analyze each system of equations:

1. y = 5x + 7
3y - 15x = 18

If the two graphs intersect at a single point, then the system has one solution. If the graphs are parallel and never intersect, then the system has no solution. If the graphs overlap each other (coincide) completely, then the system has infinitely many solutions.

2. y = 3x + 6
y = -13x - 4

Perform the same steps as mentioned above to graph the equations and determine the number of solutions.

3. x - 2y = 6
3x - 6y = 18

Again, graph the equations and analyze the intersection point(s) to determine the number of solutions.

4. y = 23x - 1
y = 23x - 2

Follow the same process as before to graph the equations and find the number of solutions.

After graphing the equations and observing the intersections, you can match the systems of equations with the appropriate number of solutions: One solution, No solution, or Infinite solutions.