Susie determined that the system of equations below has infinity many solutions.
The first like goes through the points (2,-13) and (-2,-1).
The second like goes through the points (5,-22) and (-3,2).
Is she correct?
To determine if Susie is correct, we need to check if the two lines have the same slope and y-intercept.
First, let's find the slope of the first line using the given points (2, -13) and (-2, -1). The slope formula is given by:
m = (y2 - y1) / (x2 - x1)
m = (-1 - (-13)) / (-2 - 2)
m = (-1 + 13) / (-2 - 2)
m = 12 / (-4)
m = -3
Now, let's find the slope of the second line using the given points (5, -22) and (-3, 2).
m = (2 - (-22)) / (-3 - 5)
m = (2 + 22) / (-3 - 5)
m = 24 / (-8)
m = -3
Both lines have a slope of -3, so they have the same slope.
Next, let's find the y-intercept of the first line. We can use the slope-intercept form of a linear equation, y = mx + b, where m is the slope and b is the y-intercept. We'll use the point (2, -13) and solve for b:
-13 = -3(2) + b
-13 = -6 + b
b = -13 + 6
b = -7
Now, let's find the y-intercept of the second line. We can use the slope-intercept form again, this time using the point (5, -22):
-22 = -3(5) + b
-22 = -15 + b
b = -22 + 15
b = -7
Both lines have the same y-intercept of -7.
Since the two lines have the same slope (-3) and y-intercept (-7), they are the same line. Therefore, the system of equations has infinitely many solutions. Susie is correct.
To determine if the system of equations has infinitely many solutions, we need to check if the two lines are parallel or if they are the same line.
Let's start by finding the slope of the first line using the formula:
slope = (change in y)/(change in x)
The change in y is -1 - (-13) = 12.
The change in x is -2 - 2 = -4.
So, slope of the first line = 12/(-4) = -3.
Now let's find the slope of the second line using the same formula:
The change in y is 2 - (-22) = 24.
The change in x is -3 - 5 = -8.
So, slope of the second line = 24/(-8) = -3.
Since both lines have the same slope, but different y-intercepts, they are parallel and will never intersect. Therefore, Susie's statement is incorrect.
To determine if the system of equations has infinitely many solutions, we need to check if the lines represented by the equations intersect at a single point or if they are coincident (meaning they overlap and have an infinite number of common points).
Let's start by finding the equations of the lines given the pairs of points provided.
For the first line:
- We have two points: (2, -13) and (-2, -1).
- We can find the equation of the line passing through these points using the slope-intercept form: y = mx + b.
- First, let's find the slope (m) of the line:
m = (change in y) / (change in x) = (-1 - (-13)) / (-2 - 2) = (-1 + 13) / (-2 - 2) = 12 / (-4) = -3.
- Now, we can substitute one of the points along with the slope into the equation y = mx + b to find the y-intercept (b).
- Using the point (2, -13):
-13 = -3 * 2 + b
-13 = -6 + b
b = -13 + 6
b = -7.
- Therefore, the equation for the first line is y = -3x - 7.
Now, let's find the equation for the second line:
- We have two points: (5, -22) and (-3, 2).
- Again, we can find the equation of the line using the slope-intercept form: y = mx + b.
- Finding the slope (m) of the line:
m = (change in y) / (change in x) = (2 - (-22)) / (-3 - 5) = (2 + 22) / (-3 - 5) = 24 / (-8) = -3.
- Substituting one of the points along with the slope into the equation y = mx + b to find the y-intercept (b).
- Using the point (5, -22):
-22 = -3 * 5 + b
-22 = -15 + b
b = -22 + 15
b = -7.
- Therefore, the equation for the second line is y = -3x - 7.
Comparing the equations of both lines, we can see that they are the same: y = -3x - 7. This indicates that the lines are coincident, meaning they overlap and have an infinite number of common points.
Thus, Susie is correct that the system of equations has infinitely many solutions.