Inches per feet y=12,1 y=24,1 y=36,3. The proportional relationship between inches and feet is displayed on the graph. Derive the equation of the line through the origin.
To derive the equation of the line through the origin, we need to determine the slope of the line.
The slope of a line is calculated as the change in the y-values divided by the corresponding change in the x-values between any two points on the line.
From the given data, we can select any two points to calculate the slope. Let's choose the points (1, 12) and (3, 36).
Change in y = 36 - 12 = 24
Change in x = 3 - 1 = 2
The slope (m) is calculated as the change in y divided by the change in x:
m = (Change in y) / (Change in x) = 24 / 2 = 12
Now we have the slope of the line, which is 12. The equation of a line in slope-intercept form is y = mx, where m is the slope. Since the line passes through the origin (0, 0), the y-intercept (b) is 0.
Therefore, the equation of the line through the origin is:
y = 12x
To derive the equation of the line through the origin, we need to determine the slope of the line.
The slope of a line can be found using the formula: slope = (change in y) / (change in x)
From the given data points:
- The change in x (inches) is always 12 because 12 inches make 1 foot.
- The change in y (feet) is given as follows:
- From y=12,1 to y=24,1, the change in y is 24 - 12 = 12.
- From y=24,1 to y=36,3, the change in y is 36 - 24 = 12.
Thus, the slope of the line is: slope = (12) / (12) = 1.
The equation of a line through the origin is y = mx, where m is the slope. Therefore, the equation of the line is y = 1x, or simply y = x.
To find the equation of the line through the origin that represents the proportional relationship between inches and feet, we need to calculate the slope of the line.
The slope of a line is given by the formula:
slope = change in y / change in x
In this case, the change in y represents the change in inches, and the change in x represents the change in feet.
Let's calculate the slope using the given data points:
Point 1: (y1, x1) = (12, 1)
Point 2: (y2, x2) = (24, 1)
Point 3: (y3, x3) = (36, 3)
Slope between point 1 and point 2:
slope1 = (y2 - y1) / (x2 - x1)
= (24 - 12) / (1 - 1)
= 12 / 0 (undefined)
Slope between point 2 and point 3:
slope2 = (y3 - y2) / (x3 - x2)
= (36 - 24) / (3 - 1)
= 12 / 2
= 6
Since the slope between the two data points (1, 12) and (3, 36) is consistent, we can conclude that the slope is constant.
Therefore, the equation of the line through the origin is:
y = mx
where m is the slope. In this case, m = 6.
Thus, the equation of the line is:
y = 6x