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.
To find the equation of a line that passes through two given points, we can use the point-slope form of a linear equation:
y - y1 = m(x - x1),
where (x1, y1) are the coordinates of one of the points, and m is the slope of the line.
First, let's find the slope between the two given points:
m = (y2 - y1) / (x2 - x1),
where (x1, y1) = (2/3, -5/4) and (x2, y2) = (-3, -5/6).
Substituting the values, we get:
m = [(-5/6) - (-5/4)] / (-3 - (2/3)).
Simplifying the expression:
m = (-5/6 + 5/4) / (-3 - 2/3),
m = [(10/12) - (15/12)] / (-9/3 - 2/3),
m = [-5/12] / (-11/3).
To divide fractions, we multiply by the reciprocal of the divisor:
m = [-5/12] * [-3/11],
m = 15/132,
m = 5/44.
Now that we have the slope, we can use either of the given points (2/3, -5/4 or -3, -5/6) along with the slope to find the equation of the line.
Using the point-slope form, let's use the point (2/3, -5/4):
y - (-5/4) = (5/44)(x - 2/3),
y + 5/4 = (5/44)(x - 2/3),
Now, let's simplify the equation and write it in slope-intercept form (y = mx + b) and standard form (Ax + By = C) with integer coefficients.
Slope-intercept Form:
First, multiply both sides of the equation by 44 to eliminate fractions:
44y + 55 = 5(x - 2/3),
Distribute the 5 on the right side:
44y + 55 = 5x - (10/3).
Now, move the constant term to the right side:
44y = 5x - (10/3) - 55,
44y = 5x - (10/3) - (165/3),
44y = 5x - (175/3).
Divide both sides by 44:
y = (5/44)x - (175/132).
Therefore, the equation of the line in slope-intercept form is:
y = (5/44)x - (175/132).
Standard Form:
To convert the equation to standard form, we need to eliminate fractions and express the equation in the form Ax + By = C, where A, B, and C are integers.
Multiply both sides of the equation by 132 to eliminate fractions:
132y = 132(5/44)x - (132)(175/132),
132y = 15x - 175.
Now, rearrange the terms:
15x - 132y = 175.
Therefore, the equation of the line in standard form with integer coefficients is:
15x - 132y = 175.