how do you find the equation line of these ordered pairs (1/6, -1/3) and (5/6, 5)?
A common form of a linear equation in the two variables x and y is:
y=mx+b
where m and b designate constants. The origin of the name "linear" comes from the fact that the set of solutions of such an equation forms a straight line in the plane. In this particular equation, the constant m determines the slope or gradient of that line, and the constant term "b" determines the point at which the line crosses the y-axis, otherwise known as the y-intercept.
m=(y2-y1)/(x2-x1)
b=(y1*x2-y2*x1)/(x2-x1)
In this case:
x1=1/6
x2=5/6
y1= -1/3
y2=5
m=(y2-y1)/(x2-x1)
m=[5-(-1/3)]/[(5/6)-(1/6)]
m=[5+(1/3)]/(4/6)
m=[(15/3)+(1/3)]/(4/6) Becouse 5=15/3
m(16/3)/(4/6)
m=(16*6)/(3*4)
m=96/12
m=8
b=(y1*x2-y2*x1)/(x2-x1)
b=[(-1/3)*(5/6)-((5*(1/6)]/[(5/6)-(1/6)]
b=[(-5/18)-(5/6)]/(4/6)
b=[(-5/18-(15/18)]/(4/6)
Becouse 5/6=15/18
b=(-20/18)/(4/6)
b=(-20*6)/(4*18)
b= -120/72
b=(-12*10)(12*6)
b=(-10/6)
b= -5/3 Becouse (-10/6)= -5/3
y=mx+b
y=8x-(5/3)
To find the equation of a line given two points, you can use the slope-intercept form of a linear equation, which is:
y = mx + b
where "m" represents the slope of the line, and "b" represents the y-intercept.
Step 1: Find the slope (m)
The slope (m) can be determined using the formula:
m = (y2 - y1) / (x2 - x1)
In this case, the points are (1/6, -1/3) and (5/6, 5). Let's label them as (x1, y1) and (x2, y2) respectively:
x1 = 1/6, y1 = -1/3
x2 = 5/6, y2 = 5
Substituting these values into the formula, we get:
m = (5 - (-1/3)) / (5/6 - 1/6)
m = (5 + 1/3) / (5/6)
m = (16/3) / (5/6)
m = (16/3) * (6/5)
m = 32/5 = 6.4
So, the slope of the line is 6.4.
Step 2: Find the y-intercept (b)
To find the y-intercept, we can use either of the given points. Let's use (5/6, 5):
Using the slope-intercept form, we can substitute the x-value, y-value, and slope to solve for b:
y = mx + b
5 = 6.4 (5/6) + b
5 = 8/3 + b
5 - 8/3 = b
(15/3) - (8/3) = b
7/3 = b
So, the y-intercept (b) is 7/3.
Step 3: Write the equation
Now that we have the slope (m = 6.4) and the y-intercept (b = 7/3), we can write the equation of the line:
y = mx + b
y = 6.4x + 7/3
Therefore, the equation of the line passing through the points (1/6, -1/3) and (5/6, 5) is y = 6.4x + 7/3.