What is the slope of the line that passes through the pair of points (1/2, −4) and ( − 2/3, 5) ?
The slope of the line is the ratio of "vertical change" to "horizontal change" and is denoted by the letter m.
m = ( y2 - y1 ) / ( x2 - x1 )
In this case:
x1 = 1 / 2 , y1 = - 4 , x2 = - 2 / 3 , y2 = 5
m = ( y2 - y1 ) / ( x2 - x1 )
m = [ 5 - ( - 4 ) ] / ( - 2 / 3 - 1 / 2 ) =
( 5 + 4 ) / ( - 2 ∙ 2 / 3 ∙ 2 - 1 ∙ 3 / 2 ∙ 3 ) =
9 / ( - 4 / 6 - 3 / 6 ) = 9 / ( - 7 / 6 ) = 9 ∙ 6 / - 7 = - 54 / 7
To find the slope of a line passing through two points, we can use the formula:
slope = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of the two points.
Given the points (1/2, -4) and (-2/3, 5), let's assign them as follows:
(x1, y1) = (1/2, -4)
(x2, y2) = (-2/3, 5)
Now we can substitute the values into the formula:
slope = (5 - (-4)) / (-2/3 - 1/2)
To simplify further, let's find the common denominator of the fractions:
slope = (5 - (-4)) / (-4/6 - 3/6)
= (5 + 4) / (-7/6)
= 9 / (-7/6)
To divide by a fraction, we can multiply by its reciprocal:
slope = 9 * (-6/7)
= -54/7
Therefore, the slope of the line that passes through the points (1/2, -4) and (-2/3, 5) is -54/7.
To find the slope of a line passing through two points, you can use the formula:
slope = (y₂ - y₁) / (x₂ - x₁)
Where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points.
Let's substitute the given points into the formula:
x₁ = 1/2 y₁ = -4
x₂ = -2/3 y₂ = 5
Now we can calculate the slope:
slope = (5 - (-4)) / (-2/3 - 1/2)
To simplify the calculation, we can find a common denominator for the denominators:
-2/3 = -4/6
1/2 = 3/6
Now, the equation becomes:
slope = (5 - (-4)) / (-4/6 - 3/6)
We can simplify the numerator:
5 - (-4) = 5 + 4 = 9
And the denominator:
-4/6 - 3/6 = -7/6
So, the slope is:
slope = 9 / (-7/6)
To divide by a fraction, we can multiply by its reciprocal:
slope = 9 * (−6/7)
Finally, calculating the value:
slope = -54/7
Therefore, the slope of the line passing through the points (1/2, -4) and (-2/3, 5) is -54/7.