15. At Generic High School, in the year 2001, there were 20 students per every computer. In 2008, there were 4 students per computer. Write two ordered pairs to represent the ratio of student to computer. (Let x equal the number of years since 2000.) Let x represent the time.
16. Find the slope-intercept form of the equation that goes through these two points. Must be in fraction form.
17. Find the point-slope form of the equation that goes through these two points.
You would have the points (1,20) and (8,4)
slope = (4-20)/(8-1) = -16/7 , (question could have been planned a bit better)
y - 20 = (-16/7)(x - 1)
clean it up if need be
Thank you so much!
To find the ordered pairs representing the ratio of student to computer, we can use the given information:
In the year 2001 (x = 1), the ratio of students to computers at Generic High School was 20:1. So, one pair of ordered pairs would be (1, 20).
In the year 2008 (x = 8), the ratio of students to computers at Generic High School was 4:1. So, another pair of ordered pairs would be (8, 4).
Therefore, the two ordered pairs representing the ratio of student to computer are (1, 20) and (8, 4).
To find the slope-intercept form of the equation that goes through the given two points, we can use the slope formula:
slope (m) = (y2 - y1) / (x2 - x1)
Using the ordered pairs (1, 20) and (8, 4), we get:
m = (4 - 20) / (8 - 1)
m = -16 / 7
Now, we can use the slope-intercept form of a linear equation:
y = mx + b
Using one of the given points, let's say (1, 20), we can substitute the values and solve for b:
20 = (-16/7)(1) + b
20 = -16/7 + b
b = 20 + 16/7
b = (140 + 16) / 7
b = 156/7
Therefore, the slope-intercept form of the equation that goes through the given two points is y = (-16/7)x + 156/7 (in fraction form).
To find the point-slope form of the equation that goes through the given two points, we can use the point-slope formula:
y - y1 = m(x - x1)
Using the ordered pairs (1, 20) and (8, 4), and the slope value calculated above (-16/7), we get:
y - 20 = (-16/7)(x - 1)
Expanding and simplifying:
y - 20 = (-16/7)x + 16/7
Now, we can rearrange it to get the point-slope form:
y = (-16/7)x + 16/7 + 20
y = (-16/7)x + (16 + 20 * 7) / 7
y = (-16/7)x + (16 + 140) / 7
y = (-16/7)x + 156/7
Therefore, the point-slope form of the equation that goes through the given two points is y = (-16/7)x + 156/7.
15. To calculate the ratio of students to computers, we need to determine the number of students and computers in each scenario.
In 2001, the ratio was 20 students per computer. We can represent this as the ordered pair (x, y), where x is the number of years since 2000 (x = 1 in this case) and y is the ratio of students to computers (y = 20).
In 2008, the ratio was 4 students per computer. Again, representing this as an ordered pair (x, y), where x is the number of years since 2000 (x = 8 in this case) and y is the ratio of students to computers (y = 4).
So the two ordered pairs to represent the ratio of student to computer are (1, 20) and (8, 4).
16. To find the slope-intercept form of the equation that goes through two points (x1, y1) and (x2, y2), we can use the formula:
y = mx + b
where m is the slope and b is the y-intercept.
Let's say the two points given are (x1, y1) = (x1, y1) and (x2, y2) = (x2, y2).
To find the slope (m), we can use the formula:
m = (y2 - y1) / (x2 - x1)
Substituting the values of the given points, we get:
m = (y2 - y1) / (x2 - x1)
Now, plug the slope (m) and one of the points (x1, y1) into the equation y = mx + b to find the y-intercept (b). So the equation becomes:
y1 = mx1 + b
Now isolate b:
b = y1 - mx1
So the slope-intercept form of the equation through the two points is:
y = m(x - x1) + y1. (In fraction form, y = (y2 - y1) / (x2 - x1) * (x - x1) + y1)
17. The point-slope form of an equation that goes through two points (x1, y1) and (x2, y2) is given by:
y - y1 = m(x - x1)
To find the point-slope form, first calculate the slope (m) using the formula:
m = (y2 - y1) / (x2 - x1)
Substituting the given values, we have:
m = (y2 - y1) / (x2 - x1)
Now, plug this value of the slope (m) and one of the points (x1, y1) into the point-slope equation:
y - y1 = m(x - x1)
So the point-slope form of the equation is:
y - y1 = (y2 - y1) / (x2 - x1) * (x - x1)