1) a line has a slope of 5. it passes through the points (1,4) and (6,y). what is the value of y? explain how you found your answer.
2) write a direct variation equation that relates x miles to y kilometers.
or
slope = (y-4)/5
= 5
(y-4)/5 = 5
y-4 = 25
y = 29
1) To find the value of y, we can use the slope-intercept form of a line, which is y = mx + b.
Given that the slope (m) is 5 and the line passes through the points (1,4) and (6,y), we can plug in the values to find the equation of the line.
First, we can find the value of b (the y-intercept). We substitute the coordinates (1,4) into the equation:
4 = 5(1) + b.
Simplifying the equation, we have:
4 = 5 + b.
By subtracting 5 from both sides, we find:
b = -1.
So the equation of the line is:
y = 5x - 1.
To find the value of y, we substitute x = 6 into the equation:
y = 5(6) - 1.
Calculating this, we get:
y = 30 - 1.
Therefore, y = 29.
Hence, the value of y is 29.
2) The direct variation equation that relates x miles to y kilometers can be expressed as y = kx, where k is the constant of variation.
To convert miles to kilometers, we need to know the conversion factor: 1 mile is equal to approximately 1.60934 kilometers.
So, to write the direct variation equation, we can substitute the conversion factor (k = 1.60934) into the equation:
y = 1.60934x.
Therefore, the direct variation equation that relates x miles to y kilometers is y = 1.60934x.
1) To find the value of y, we need to use the point-slope formula. The formula for a line with slope m passing through the point (x₁, y₁) is y - y₁ = m(x - x₁).
Given that the line has a slope of 5 and passes through the points (1,4) and (6,y), we can substitute the values into the point-slope formula. Let's start with the first point (1,4):
y - 4 = 5(x - 1)
Now, let's solve for y by substituting the second point (6,y):
y - 4 = 5(6 - 1)
Simplifying, we have:
y - 4 = 5(5)
y - 4 = 25
Finally, to isolate y, we add 4 to both sides of the equation:
y = 25 + 4
y = 29
Therefore, the value of y is 29.
2) In a direct variation equation, y varies directly with x, which means y and x are proportional. The general form of a direct variation equation is y = kx, where k is the constant of variation.
To relate x miles to y kilometers, we need to establish the conversion factor between miles and kilometers. Let's say there are 1.60934 kilometers in a mile.
Using this conversion factor, we can write the direct variation equation:
y = 1.60934x
This equation relates x miles to y kilometers, as y is directly proportional to x with the constant of variation being the conversion factor.
on the 1,4 point
y=mx+b
4=5*1+b
b=-1
on the 6,y point
y=5*6-1=29