Susie determined that the system of equations below has infinitely many solutions. %0D%0A%0D%0AThe first line goes through the points (2, -13) and (-2, -1).%0D%0A%0D%0AThe second line goes through the points (5, -22) and (-3, 2).%0D%0A%0D%0AIs she correct? *Use your Desmos calculator%0D%0A%0D%0A(1 point)
To determine whether the system of equations has infinitely many solutions, we can graph the equations and see if the lines intersect at more than one point.
For the first line, we can use the two given points to find its equation.
The slope of the line is {(y2-y1)/(x2-x1) = (-1-(-13))/(-2-2) = 12/4 = 3}.
Using the point-slope form of a line, we can write the equation as:
y - y1 = m(x - x1),
where m is the slope and (x1, y1) is any point on the line.
Using the point (2, -13), the equation can be written as:
y + 13 = 3(x - 2),
which simplifies to:
y = 3x - 6 - 13,
y = 3x - 19.
For the second line, we can again use the two given points to find its equation.
The slope of the line is {(y2-y1)/(x2-x1) = (2-(-22))/(-3-5) = 24/(-8) = -3}.
Using the point-slope form of a line, we can write the equation as:
y - y1 = m(x - x1).
Using the point (5, -22), the equation can be written as:
y + 22 = -3(x - 5),
which simplifies to:
y = -3x + 15 - 22,
y = -3x - 7.
Now we can graph the two equations on the Desmos calculator to see if they intersect at more than one point.
Based on the graph, we can see that the two lines intersect at exactly one point (-1, -4), indicating that the system of equations does not have infinitely many solutions.
Therefore, Susie is incorrect.
To check if Susie is correct, let's plot the given points and lines on the Desmos calculator and see if they intersect at a single point or not.
First, let's plot the points (2, -13), (-2, -1), (5, -22), and (-3, 2).
Now, let's plot the first line that goes through the points (2, -13) and (-2, -1).
To do this, we can use the slope-intercept form of a line, which is y = mx + b, where m is the slope and b is the y-intercept.
The slope of the line can be found using the formula: slope (m) = (y2 - y1) / (x2 - x1), substituting the coordinates of the two points we have, we get:
m = (-1 - (-13)) / (-2 - 2) = 12 / -4 = -3
To find the y-intercept (b), we can substitute the coordinates of one of the points into the equation y = mx + b. Let's use the point (2, -13):
-13 = -3(2) + b
-13 = -6 + b
b = -13 + 6
b = -7
Therefore, the equation of the first line is y = -3x - 7.
Now, let's plot the second line that goes through the points (5, -22) and (-3, 2).
Similarly, calculating the slope:
m = (2 - (-22)) / (-3 - 5) = 24 / -8 = -3
Calculating the y-intercept using the point (5, -22):
-22 = -3(5) + b
-22 = -15 + b
b = -22 + 15
b = -7
Therefore, the equation of the second line is y = -3x - 7.
Now, we can observe the Desmos graph and see if the lines intersect at a single point or not. If they intersect at a single point, Susie's statement that the system of equations has infinitely many solutions would be incorrect.
After analyzing the graph, we can conclude that the lines do not intersect at a single point, but instead overlap completely. This means that the system of equations has infinitely many solutions.
So, Susie is correct.
To determine if Susie's statement is correct, we can use the Desmos calculator. Here's how you can do it:
1. Open the Desmos graphing calculator in a web browser or install the Desmos app on your device.
2. On the calculator screen, you'll see a coordinate plane. Click on the "+" button in the top-left corner of the calculator to add new lines.
3. Let's start with the first equation, which goes through the points (2, -13) and (-2, -1). To enter this equation, type "y = mx + b" in the input bar at the top of the calculator, replacing "m" with the slope and "b" with the y-intercept. The slope can be calculated using the formula (change in y)/(change in x).
- For the first equation, the slope is (-13 - (-1))/(2 - (-2)) = -12/4 = -3. So, the equation becomes "y = -3x + b".
- To find the value of the y-intercept "b," substitute the coordinates of either point into the equation. Let's use (2, -13):
- -13 = -3(2) + b
- -13 = -6 + b
- b = -13 + 6
- b = -7
- Therefore, the first equation is "y = -3x - 7". Enter this equation into Desmos.
4. Now, let's move on to the second equation, which goes through the points (5, -22) and (-3, 2). Follow the same steps as before:
- Calculate the slope: (-22 - 2)/(5 - (-3)) = -24/8 = -3. The equation becomes "y = -3x + b".
- Substitute the coordinates of either point, let's use (5, -22):
- -22 = -3(5) + b
- -22 = -15 + b
- b = -22 + 15
- b = -7
- Therefore, the second equation is "y = -3x - 7". Enter this equation into Desmos.
5. Once both equations are entered, Desmos will plot the lines on the coordinate plane. If the lines coincide or overlap, it means there are infinitely many solutions, and Susie is correct. If the lines are distinct and do not overlap, it means there is a unique solution, and Susie is incorrect.
6. Compare the lines on the graph to determine if they coincide.
By following these steps and using the Desmos calculator, you can check if Susie's statement about the system of equations having infinitely many solutions is correct.