Find equation of the line that passes through the points (2/3,-5/4) and (-3,-5/6). write your answer in slope-intercept form and standard form with integer coefficients.
The slope m of a line passing through two points (x1, y1), (x2, y2) is given by
m = (y2-y1)/(x2-x1)
m = (-5/6 - -5/4) / (-3-2/3) = (5/12) / (-11/3) = -15/132 = -5/44
The equation for a straight line is of the form y = m*x + b
where m is the slope and b is the y-intercept.
So far we have;
y = -5/44*x + b
To solve for b, plug in one of the points:
-5/4 = -5/44 * (2/3) + b
-5/4 = -10/132 + b
b = -5/4 + 10/132 = -155/132
y = -5/44*x - 155/132 (slope-intercept form)
Standard form: You need to find the common factors of 44 and 132: it's 132, so multiply the equation by 132:
132*y = -15*x - 155
-15*x + 132*y = -155
To find the equation of a line that passes through two given points, we can use the slope-intercept form of a linear equation: y = mx + b, where m is the slope of the line and b is the y-intercept.
Step 1: Finding the slope (m)
The slope (m) can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
Let's label the first point as (x1, y1) = (2/3, -5/4) and the second point as (x2, y2) = (-3, -5/6).
Substituting these values into the slope formula:
m = ((-5/6) - (-5/4)) / (-3 - (2/3))
To simplify the calculations, let's find a common denominator. The common denominator for 6 and 4 is 12.
m = ((-5/6) - (-5/4)) / (-3 - (2/3))
= (-5/6 + 5/4) / (-3 - (2/3))
= (-10/12 + 15/12) / (-9/3 - 2/3)
= (5/12) / (-7/3)
= (5/12) * (-3/7)
= -15/84
= -5/28
So, the slope (m) of the line passing through the given points is -5/28.
Step 2: Finding the y-intercept (b)
To find the y-intercept (b), we can substitute the coordinates of one of the points (2/3, -5/4) into the slope-intercept form (y = mx + b) and solve for b.
Using the point (2/3, -5/4):
-5/4 = (-5/28)(2/3) + b
Let's simplify the right-hand side:
-5/4 = (-5/14) + b
Adding 5/14 to both sides:
-5/4 + 5/14 = b
-35/28 + 10/28 = b
-25/28 = b
So, the y-intercept (b) is -25/28.
Step 3: Writing the equation in slope-intercept form
Now that we have the slope (m = -5/28) and the y-intercept (b = -25/28), we can write the equation in slope-intercept form (y = mx + b):
y = (-5/28)x - 25/28
Step 4: Writing the equation in standard form with integer coefficients
To convert the equation to standard form with integer coefficients, we can multiply the entire equation by 28 to eliminate the fractions:
28y = -5x - 25
Now, rearranging the equation so that the coefficients are integers:
5x + 28y = -25
Therefore, the equation of the line passing through the points (2/3, -5/4) and (-3, -5/6) in slope-intercept form is y = (-5/28)x - 25/28, and in standard form with integer coefficients is 5x + 28y = -25.