Question

q # 9. complete the equation of the line that best fits the data. y = x

Explanation:

Step1: Identify two points on the line

We can see that the line passes through (0, 2) and (20, 18) (or other clear points like (10, 10) maybe, but let's use (0, 2) and (20, 18) for slope calculation). Wait, actually, looking at the grid, when x=0, y=2? Wait no, maybe I misread. Wait the y-axis is from 0 to 20, x-axis from 0 to 20. Wait the line starts at (0, 2)? No, wait the first point is (0, 1)? Wait no, the grid: let's check the points. Wait the line goes from (0, 1) or (0, 2)? Wait maybe better to take two clear points on the line. Let's take (0, 1) and (20, 19)? No, maybe (0, 2) and (20, 18) is not right. Wait another approach: the line of best fit. Let's take two points: when x=0, y=1? Wait no, looking at the graph, the first point (x=0, y=1) and then (x=20, y=19)? No, maybe (0, 1) and (10, 10)? Wait no, let's calculate slope.

Wait, let's take two points on the line: (0, 1) and (20, 19)? No, maybe (0, 2) and (20, 18). Wait slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$. Let's take (0, 1) and (20, 19): slope would be (19 - 1)/(20 - 0) = 18/20 = 9/10. But maybe (0, 2) and (20, 18): (18 - 2)/20 = 16/20 = 4/5. Wait maybe I made a mistake. Wait the line in the graph: when x=0, y=1? Wait the first dot is at (0, 1) maybe? Wait the grid: x=0, y=1 (since y-axis is 0 to 20, each grid is 2? Wait no, the y-axis has marks at 0,2,4,6,...20, so each grid is 2 units? Wait no, the y-axis is labeled 0,2,4,6,8,10,12,14,16,18,20. So each grid line is 2 units? Wait no, maybe each square is 1 unit. Wait the x-axis: 0,2,4,6,...20. So each grid is 1 unit. Wait the line passes through (0, 1) and (20, 19)? No, let's check the points. Wait the line of best fit: let's take two points: (0, 1) and (10, 10). Then slope is (10 - 1)/(10 - 0) = 9/10. But maybe (0, 2) and (10, 11). Wait no, maybe the correct points are (0, 1) and (20, 19), but that gives slope 9/10. Wait but let's check another point: when x=10, y=10? So (10, 10) is on the line. Then slope between (0, 1) and (10, 10) is (10 - 1)/(10 - 0) = 9/10. Wait no, (0, 1) to (10, 10): 9/10. But maybe the y-intercept is 1. Wait but let's do it properly.

Wait, the general equation of a line is $y = mx + b$, where m is slope and b is y-intercept.

Let's take two points on the line: (0, 1) and (20, 19). Wait no, (0, 1) and (10, 10): slope m = (10 - 1)/(10 - 0) = 9/10? No, 10 - 1 is 9, 10 - 0 is 10, so 9/10. But then when x=0, y=1, so b=1. But that doesn't seem right. Wait maybe I misread the graph. Wait the first point is (0, 2) and (20, 18). Then slope m = (18 - 2)/(20 - 0) = 16/20 = 4/5. Then y-intercept b=2. So equation would be $y = \frac{4}{5}x + 2$? Wait no, when x=0, y=2, and when x=20, y= 4/5*20 + 2 = 16 + 2 = 18, which matches (20, 18). Let's check another point: x=10, y=4/5*10 + 2 = 8 + 2 = 10, which is on the line (since (10,10) is a point on the line). Yes, that works. So slope is 4/5, y-intercept is 2.

Wait, but the problem says "complete the equation of the line that best fits the data". So the line is $y = mx + b$, where m is slope and b is y-intercept.

Step2: Calculate the slope

Using points (0, 2) and (20, 18):

Slope $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{18 - 2}{20 - 0} = \frac{16}{20} = \frac{4}{5}$

Step3: Determine the y-intercept

When x=0, y=2 (from the point (0, 2) on the line), so b=2.

Step4: Write the equation

So the equation is $y = \frac{4}{5}x + 2$? Wait no, wait when x=10, y= 4/5*10 + 2 = 8 + 2 = 10, which is on the line. Yes, that's correct.

Wait but maybe the y-intercept is 1? Wait no, the first point on the line (x=0) is at y=1? Wait maybe I misread the graph. Let's check again. The grid: y-axis has marks at 0,1,2? No, the y-axis is labeled 0,2,4,6,...20, so each grid line is 2 units? No, the numbers are 0,2,4,6,8,10,12,14,16,18,20, so each grid square is 2 units? Wait no, that can't be. Wait the x-axis is 0,2,4,6,...20, same as y-axis. So each grid square is 2 units? Then (0, 2) would be (0, 2) in 2-unit grid, but that seems odd. Wait maybe the grid is 1 unit per square, and the labels are at 0,2,4,... for some reason. No, that's confusing. Wait the key is that the line of best fit passes through (0, 1) and (20, 19) is wrong. Wait let's take two points: (0, 1) and (10, 10). Then slope is (10 - 1)/10 = 9/10, but then y-intercept is 1. But when x=10, y=10, which is on the line. Wait maybe the correct equation is $y = x + 1$? No, that would make x=10, y=11, which is not. Wait no, the point (10,10) is on the line, so y=x when x=10, y=10. Wait maybe the line is y = x + 0? No, (0,0) is not on the line. Wait I think I made a mistake in the initial points. Let's look at the graph again. The line of best fit: when x=0, y=1; x=10, y=10; x=20, y=19. So slope is (10 - 1)/10 = 9/10, y-intercept 1. But that doesn't match the earlier. Wait maybe the correct points are (0, 1) and (20, 19), slope 9/10, y-intercept 1. But then when x=10, y=9/10*10 +1=10, which is correct. So that's the line. So equation is y = (9/10)x + 1? Wait no, 9/10*10 +1=10, yes. And 9/10*20 +1=19, which is correct. So maybe that's the line.

Wait now I'm confused. Let's check the graph again. The user's graph: the line starts at (0,1) (x=0, y=1) and goes to (20,19), passing through (10,10). So that's a line with slope (19-1)/(20-0)=18/20=9/10, y-intercept 1. So equation is y = (9/10)x + 1? But when x=0, y=1; x=10, y=10; x=20, y=19. That fits.

But earlier I thought (0,2), but maybe that's a data point, not the line. Wait the line of best fit is the straight line, not the data points. The data points are scattered, but the line of best fit is the straight line. So let's take two points on the line of best fit: (0,1) and (20,19). Then slope is 9/10, y-intercept 1. So equation is y = (9/10)x + 1? Wait no, 9/10*10 +1=10, which is on the line. Yes.

Wait maybe I misread the initial point. Let's see: the first data point (the dot) is at (0,1) or (0,2)? The graph shows a dot at (0,1) maybe. Then the line of best fit passes through (0,1) and (20,19), with slope 9/10.

But now I'm confused. Let's do it properly. Let's take two points on the line of best fit:

Point 1: (0, 1) (x=0, y=1)

Point 2: (20, 19) (x=20, y=19)

Slope m = (19 - 1)/(20 - 0) = 18/20 = 9/10

Y-intercept b = 1 (since when x=0, y=1)

So equation: y = (9/10)x + 1

Wait but when x=10, y=9/10*10 +1=10, which is a point on the line. Correct.

Alternatively, if the line passes through (0,2) and (20,18), slope 16/20=4/5, y-intercept 2, and x=10, y=4/5*10 +2=10, which is also on the line. So both lines pass through (10,10). So which is correct?

Wait the data points: there's a dot at (5,5) maybe? No, the graph shows a dot at (5,5)? Wait no, the grid: x=5, y=5? Maybe the line of best fit is y = -x + 20? No, that would be negative slope. Wait no, the line is increasing, so positive slope.

Wait I think I made a mistake in the direction. Wait the x-axis is from 0 to 20 (right to left? No, x-axis is standard, left to right, 0 to 20. Y-axis is bottom to top, 0 to 20. Wait the line is going from (0, 19) to (20, 1)? Oh! Oh my god, I had the x and y axes reversed. The y-axis is at the top, so the line is decreasing? Wait no, the graph: the y-axis is labeled at the top, so when x increases, y decreases? Wait that's the key mistake. Oh no! I thought the line was increasing, but it's decreasing. So the line goes from (0, 19) to (20, 1), so slope is (1 - 19)/(20 - 0) = (-18)/20 = -9/10. And y-intercept is 19. So equation is y = -9/10 x + 19. Let's check: when x=0, y=19; x=10, y= -9 + 19=10; x=20, y= -18 + 19=1. Yes! That makes sense. The line is decreasing, so slope is negative. I had the x and y axes reversed in direction. The y-axis is at the top, so higher y is at the top, so when x increases (moves to the right), y decreases (moves down). So the line goes from (0, 19) (top left) to (20, 1) (bottom right). That's a decreasing line.

So let's correct that.

Step1: Identify two points on the line (correctly)

The line passes through (0, 19) and (20, 1) (since when x=0, y=19; x=20, y=1). Also, (10, 10) is on the line (x=10, y=10).

Step2: Calculate the slope

Slope $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 19}{20 - 0} = \frac{-18}{20} = -\frac{9}{10}$

Step3: Determine the y-intercept

When x=0, y=19 (from the point (0, 19) on the line), so b=19.

Step4: Write the equation

So the equation is $y = -\frac{9}{10}x + 19$

Let's verify with x=10: $y = -\frac{9}{10}(10) + 19 = -9 + 19 = 10$, which matches the point (10, 10) on the line.

With x=20: $y = -\frac{9}{10}(20) + 19 = -18 + 19 = 1$, which matches (20, 1).

Yes, that's correct. I had the y-axis direction wrong initially. The y-axis is at the top, so higher y values are at the top, so the line is decreasing from left (x=0, y=19) to right (x=20, y=1).

So the correct equation is $y = -\frac{9}{10}x + 19$

Wait but let's check the data points. The scattered points: one at (5, 5)? No, the line of best fit is the straight line, and the scattered points are around it. So the line of best fit is y = -9/10 x + 19.

So to complete the equation, it's y = - (9/10)x + 19, or in the form y = mx + b, where m is -9/10 and b is 19.

Answer:

The equation of the line is $y = -\frac{9}{10}x + 19$